You didn’t know, possibly, that when he thought we was dying, he showed the notebook to one of his colleagues, who copied out the proof:

In an unsanitized, politically incorrect (but factual) history, Selmer Bringsjord talks about how the tormented genius Kurt Gödel took up a quest that dated back a thousand years to prove the existence of God by formal logic. His original version didn’t quite work but his editor’s version passed an important logic test:

“When we go to Gödel, we skip over the modern advocates of this argument. It’s harsh—I’m just going to say it—from the standpoint of someone who’s reasonably well-versed in formal logic, I think it’s a bit of a doldrums, despite some of the attention, until Gödel does his thing.

Gödel does it formally and then some folks in Germany, doing automated reasoning, verified it a few years back. They verified the version that Dana Scott copied out of the notebook. That is, what they verify is that there is no doubt; it’s machine-verified proof. So now we’re left with just the truth of the premises and how we judge them.”

News, “Gödel and God: A surprising history” atMind Matters News

*Further reading:*

Faith is the most fundamental of the mathematical tools: An early twentieth century clash of giants showed that even mathematics depends on some unprovable assumptions. (Daniel Andrés Díaz-Pachón)

and

God’s existence is proven by science. Arguments for God’s existence can be demonstrated by the ordinary method of scientific inference. *(Michael Egnor)*

Genesis 22:9-12

Although I now regard myself as agnostic/atheist, I was raised a Christian and I genuinely believed. I was taught that the true Christian needs only faith, not science or mathematics, just faith.

Later, I became aware of the contradictions in the Biblical accounts, such as how could an omniscient God possibly not know what was in Abraham’s heart, the real strength of his faith? So what purpose could this immoral test possibly serve?

^^^^^^^^ Ironic,

Seversky accuses Christians of having blind faith and no reason, yet he himself has abandoned all reason in his blind faith towards atheism.

As I pointed out the other day, the fact that Atheism itself cannot ground reasoning is powerful evidence against Atheism being true.,,, In short, “anyone who believes in reason must also believe in God.”

https://uncommondescent.com/intelligent-design/when-beliefs-dont-depend-on-reason/#comment-700669

Conclusion of preceding link,,,

@ Seversky, Your perception of god is only contradictory because it is extremely limited in the first place.

Unlimited capacity to do anything with no ability to control or choose to do so

That appears to be the standard you hold God. This is not God

Seem to be hung up on a question that I asked my parents when I was 4

Can god create a rock that God cannot lift?

If the answer is yes then God can’t do everything

If the answer is no then God can’t do everything

Seems that you’re hung up on if God knows everything then etc…..

If Seversky would take time to understand what Abraham had been raised around, he would know that child sacrifice happened with all the deities worshiped. It was not a lack of understanding on God’s part, but Abraham’s. By stopping Abraham at the moment just before sacrifice, it was made clear to him just how different God was from the other deities in the region.

Most who claim to be atheists are not atheists. You do not give human characteristics to something that does not exist. You only do so out of anger over something and desire to blame God.

Without God, there can be no life. There is not one hypothesis that comes close to satisfying the question of origin when God is removed. It is only through the inclusion of intelligence far beyond human understanding that you can get life from no life.

For Pete’s sake get a new tune, Seversky. You’ve sung that one half a dozen times.

Sev,

fideism is a strawman caricature of well founded Christian or even wider theistic faith. That said, given the Agrippa trilemma, the roots of our worldviews cannot be turtles all the way down and cannot be turtles in a circle, so the last turtle has to stand somewhere, at finite remove.

Our challenge is to have reasonable, responsible faith-points when we reach that root level, averting question-begging through comparative difficulties analysis. This requires facing the fact that all worldviews bristle with difficulties and so our challenge is to compare, across factual adequacy, coherence and balanced explanatory power. The last points to grand inference to the best explanation, the opposite of a demonstration on generally accepted axiomatic presuppositions. Mind you, there are self-evident truths that serve as plumb-line tests, but these cannot be enough to compose the framework for a worldview.

In that context, it is reasonable to ask about warrant [notice, much wider than “proof”] for belief in God.

For many, that is settled by living, life-transforming encounter. The Christian Faith points to this, in the context that the world without and our inner light of life within are adequate warrant to trust in God. That inner light in crucial part pivots on our moral government — the inescapable first duties of responsible reason I keep pointing out. Duties, that must truly be duties, on pain of utter incoherence and chaos in our inner lives: to truth, to right reason, to prudence [so, inter alia, warrant], to sound conscience, to neighbour, to fairness and justice etc. Your own arguments implicitly rest on our knowing these duties.

That such creatures [us] exist, constrains plausible candidates for the required, finitely remote world root. It must not only be causally adequate and temporally plausible but adequate to found moral government. Post-Hume, that requires bridging the IS-OUGHT gap, being inherently good.

Causal-temporal adequacy surfaces a form of the trilemma. We cannot traverse an actually transfinite temporal-causal succession of finite stages in steps, as can be seen through first focussing the relevant set for numbers, the hyperreals R*, not R, where we can see Z* as integer mileposts. (The definition of R is inadequate for what our teachers do with the algebraic number line in high school, much less other things.)

For a supposed transfinitely remote past there has to be a stage K’ [= -K in Z*] that would be actual past but is transfinitely remote; otherwise we are simply discussing finitely remote bpast points. K’, K’+1, K’+2 etc to K’+k goes on K’+k+1 etc, i.e. we can only successively advance to a finite span from K’; we never bridge to a finite span from 0 [take that as the singularity], much less a count forward to n, now. We are only

warrantedto speak of a finite past. Turtles all the way down fails. The same basic challenge extends to an infinite regress of warrant; our rationality is inherently finite and bounded.We need to pause on logic of being, recognising that across the span of possible worlds, there are candidates impossible vs possible of being, the former due to the sort of contradictory core characteristics in say a square circle. Of possibles, contingent beings would or do exist in at least one possible world [but not in others]. Relevant factors tied to that are causes for such a being c. By contrast, necessary beings, n, exist in every possible world; i.e. they are framework elements for any possible world. (Try to imagine a world in which 2 does not exist or can cease from being.)

Many of these things are now unfamiliar (thanks to the spirit of our age and what we typically deem worthy of being studied), but that does not make them irrelevant. Ontology is foundational.

In this context, we need a finitely remote, necessary being root of reality causally adequate for a fine tuned cosmos hosting a world of cell based life that includes inescapably morally governed creatures with a built in natural law attested to by sound conscience. That requires inherent goodness thus also utter wisdom, simply to bridge the IS-OUGHT gap, where also necessary being is eternal. Were there ever utter non-being such would forever obtain, so if a world is, something — the necessary being reality root — always was.

That’s a pretty stiff bill to fill, and it drastically shifts our evaluation of the warrant for theism and the various forms of theistic argument.

Indeed, after centuries of debates, there is just one serious candidate to fill the bill: the inherently good, utterly wise creator God, a necessary and maximally great being. One, worthy of loyalty and of the reasonable, responsible service of doing the good that accords with our evident nature. A familiar figure (and definitely not the sort of scapegoat sketched out in Mr Dawkins’ strawman caricature that is definitive of the new atheism, so called).

This framework also spans the picture studied in philosophical and systematic theology.

It should change, too, how we look at the plausibility of the Godel-style, modal logic ontological argument. Even, the Anselm form, suitably adjusted if needs be.

BTW, as for your struggle with Gen 22, has it registered that even with the evening prayer by one’s bedside, God is not unaware of our needs before we articulate them and indeed beyond what we can put in words? So, prayer, faith, adventures of life cannot tell God something novel? Instead, our prayers, struggles and adventures are relational, tied to soul-making constrained by where we are. God can only give us safely what we have been opened up to handle. And in this case, an obvious element is to break the idolatrous pattern of trying to manipulate gods by child sacrifice.

Though, in a sobering aspect, this points to the dilemma many a father and mother have confronted ever since: sending their sons into battle with a significant chance or even effective certainty that they will not come back. For, we live in a world of aggressive, violent, oppressive evil that sometimes has to be fought. Even, in tearing battles of attrition such as the Heurtgen Forest (there were dams that if blown would flood valleys blocking the advance), or Omaha Beach (required to bridge Utah on the Contentin peninsula with the UK-Canadian beaches and to take out key heavy gun batteries), or with the parachute divisions landing behind those five immortal beaches, or Stalingrad (the huge casualty figures of the European fronts of WW2 were in the East, right up to the battle for Berlin . . . the Russians lost nearly as many on the battle field as were lost in battle on all sides in WW1), or Verdun, or the Somme (an unready army had to take up a bleeding offensive to relieve pressure at Verdun), or Gettysburg or any number of other places and times.

I recall, then General Petain, watching 18 year old boys marching to the front at Verdun, his heart lurching.

I think some re-thinking is in order.

KF

PS: Rom 1:

Note here, Locke:

After reading the biographical bits, I’m struck by a metaquestion. If God exists, would he want his existence to be proved by a man who was so totally removed from the REALITY of God’s works that he couldn’t trust food? Godel lived in a universe of total infinite abstraction and symbolism, and handled those symbols better than anyone. Natural Law says a lot about the importance and blessings of food, but disdains overabstraction.

polistra, Godel was an eccentric genius with serious struggles. One of many, Newton springs to mind. KF

As to the ontological argument for God, Dr. Craig has made a video that explains the basics of the ontological argument for God in an easy to understand manner,, (much easier to understand than Godel’s proof which basically “looks like hieroglyphics” for the average layman):

Here are the basic steps of the ontological argument as laid out by Dr. Craig:

And as Dr. Craig points out in the following article, “it might be a surprise to learn that steps 2–7 of this argument are relatively uncontroversial. Most philosophers would agree that if God’s existence is even possible, then he must exist. So the whole question is: Is God’s existence possible? The atheist has to maintain that it’s impossible that God exists.”

And as Dr. Craig states in the following video, “It (the ontological argument) puts the atheist in a very awkward position. The atheist must deny, not merely that God exists, he must maintain that it is impossible that God exists. And that is certainly a radical claim that would require great justification.”

Where this gets VERY interesting is that atheists, in their appeal to an infinity of other possible worlds to try to ‘explain away’ the fine tuning of this universe,,,

,,, in their appeal to a multiverse to try to ‘explain away’ the fine tuning of this universe, atheists have, basically, completely conceded the necessary premise, i.e. that it is possible that God exists, to the ontological argument in order for the ontological argument to work.

Simply put, you cannot argue that it is possible that an infinity of other universes exist in which an infinity of other possibilities are playing out, while at the same time holding that it is impossible for a maximally great being to exist in one of those infinity of other universes:

The absurdity of the atheist’s predicament with his appeal to the multiverse is humorously laid out in the following article,

Thus once again, in another twist of atheistic irony, the atheist ends up shooting himself in the foot in his appeal to a multiverse in order to try to ‘explain away’ the fine-tuning of the universe and therefore avoid the implication of God, since his appeal to a multiverse, in and of itself, concedes the necessary premise to the ontological argument, i.e. that it is possible for God to exist, in order for the ontological argument to work.

Of supplemental note, I touched upon Godel’s incompleteness theorems yesterday in the following post:

BA77, in short the issue pivots on whether there is a serious candidate necessary being with key attributes that would lead to identification with the God of theism. Where, a being that exists in all possible worlds is a necessary not a contingent being. KF

PS: It’s math-roglyphics! A language all to itself, here modal logic.

PPS: A reason to believe God is impossible of being is __________, or, that he is not a serious candidate necessary being is __________ That is going to be fun to watch.

Sev,

Sorry if you were taught that sort of faith. It sounds a lot like fideism, which I rejected as soon as I found out what it was. I try to hold beliefs that are based on evidence and (hopefully sound) inference.

I was taught – and I think a lot of Christians believe – that fideism is a central principle of the faith as exemplified in the story of Doubting Thomas – “Jesus saith unto him, Thomas, because thou hast seen me, thou hast believed: blessed [are] they that have not seen, and [yet] have believed” – and in the story of the Temptation of Christ – “Again it is written, ‘You shall not put the Lord, your God, to the test.”

Bornagain77@ 10Okay, let’s take a closer look at Craig’s ontological argument.

First, what does he mean by “maximally great being”? Strongest? Fastest? Biggest? Baddest? All of the preceding and more? The argument doesn’t say so the term is undefined and essentially meaningless. So let’s replace “maximally great being” with another meaningless term borrowed from Lewis Carroll “slithy tove” and see how that works:

1. It is possible that a slithy tove exists.

2. If it is possible that a slithy tove exists, then a slithy tove exists in some possible world.

3. If a slithy tove exists in some possible world, then it exists in every possible world.

4. If a slithy tove exists in every possible world, then it exists in the actual world.

5. If a slithy tove exists in the actual world, then a slithy tove exists.

6. Therefore, a slithy tove exists

So, have we proven that a fictional creature from a nonsense poem by Lewis Carroll actually exists, that “Jabberwocky” is a documentary?

KF

There is no reason to believe that a god is impossible, just that there is no compelling evidence that one exists. And, since there is no proof that a necessary being is required, why would I feel the need to fill that unnecessary gap with a non-existent god?

Sev

“I was taught – and I think a lot of Christians believe – that fideism is a central principle of the faith “

I can’t speak as to what you were taught but I know a lot of Christians and fideism is NOT a central principle of faith. I don’t know what Christian world you were apart of but it’s not the world I am familiar with.

Vivid

@ 15 cute Parallel, misses the point completely, but it’s a cute parallel

EG, are you aware that reality requires a necessary being world root (tied to part of reality being a temporal-causal, cumulative order)? Further, that reality contains morally governed creatures (on pain of utterly undermining our rationality)? That such constraints provide a bill of requisites for such a root? Thus, the real issue is serious candidates to be such? Where, despite your report on your triumphalistic, confident skepticism — known to have hyper tendencies — God is indeed such a serious candidate? Where also, a serious candidate NB either is as part of framework for any possible world, or else is not as impossible of being, in ways analogous to say, square circles (irremediable contradictions of core proposed characteristics)? Kindly, then, provide a more responsive response. KF

PS: One of the major intellectual failings of our time has been the substitution of skepticism imagined as a virtue in the place of sound prudence.

Sev, is a slithy tove a serious candidate necessary being, especially a necessary being reality root where at least one actual world has in it morally governed creatures? On what grounds? If not, you have conflated contingent and necessary beings. If so, then perhaps “slithy tove” is an unintentional synonym for God, considered as the inherently good and utterly wise one, creator and sustainer of all worlds, a necessary and maximally great being, worthy of loyalty and of the responsible, reasonable service of doing the good that accords with our evident nature. KF

Sev, the fideism you report is distressingly common but is a grave error. In fact, the NT word for faith — pistis — significantly overlaps with a root sense: conviction by persuasive and cogent warrant, AKA rhetorical proof . . . what we can access on matters that in significant part are about on the ground facts, many being historical. And it also overlaps with fidelity in the sense loyalty and steadfastness in the face of challenges. The Christian faith addresses evidence of the world without and the inner world within, pivoting on the prophesied, fulfilled resurrection of Jesus, with 500 witnesses. KF

At 15 Seversky states,

And yet, contrary to Seversky’s assertion, the term is not ‘essentially meaningless’,

i.e. For Seversky to prove that the term ‘maximally great being’ is ‘essentially meaningless’, he must first prove that things cannot be ordered from least to greatest. Yet everyone believes that things can be hierarchically ordered in such a way from least to greatest.

We do such ordering and ranking all the time. Seversky himself does it all the time. He gave a rough outline of such a hierarchical ranking system in his post for crying out loud.

Thus Seversky, since the burden is on you to prove that the term is meaningless, and yet you yourself, in your everyday life, do not actually believe that the term is meaningless, then good luck actually proving (rather than just asserting) that the term ‘maximally great Being’ is completely meaningless.

Might it also be too obvious to point out the fact that Lewis Carroll’s “slithy tove” ranks VERY low in terms of any hierarchy that would be used to judge whether a Being was actually maximally great or not? Indeed, I can find much that is NOT maximally great with the fictional creature:

of footnote: It might also be good for Seversky to first prove that he actually exists as real person, (and that he not just a ‘neuronal illusion’), before he then tries to argue against the reality of God as a ‘maximally great’ person.

BA77, excellent, so slithy tove is at best a contingent being. On that, Sev’s comparative collapses by way of failure to recognise modes of being and linked logic of being. Linked, not only is greatness routinely recognised as present among beings in varying degree, but such is real: a Hitler is lesser than a Churchill, never mind the latter’s great flaws. In Cricket, the all conquering Windies of ~ 40 years ago are far beyond the pale shadows we see today. Passing onward, we can understand great-making properties, especially virtues. A maximally great being possesses only such properties and all relevant properties to maximal compossible degree: the supreme and supremely good and wise one. Maximal greatness is far from meaningless, as can be seen in the sketched out summary I gave above, starting with inherently good and utterly wise, as well as reality root. KF

PS: Sev’s error is directly parallel with rhetoric about flying spaghetti monsters; which has been seriously put up by many current advocates of atheism. That is how ill-informed we are today.

Kairosfocus: are you aware that reality requires a necessary being world root (tied to part of reality being a temporal-causal, cumulative order)?Sorry, sorry, I’ve been just lurking on this thread but I must ask: what is a necessary being world root? I’m missing some point of theology obviously.

I was raised a Christian and went to Catholic schools. I was NEVER taught that Christians didn’t need science and math, Quite the opposite. We were taught that science and math are our way of understanding God’s Creation. We were taught that after the Fall we had to learn to fend for ourselves and tat math and science were the way to do so.

As for Abraham, perhaps God saw that he was in conflict. And by having him sacrifice his child that conflict would be resolved.

@25

The same, never faith alone, science was to understand the creation, and it was faith with acts

@24

A Progenitor being, something where all things stem from, source of the beginning, this is what I believe he means by that, correct me if I’m wrong KF

AaronS1978: A Progenitor being, something where all things stem from, source of the beginning, this is what I believe he means by that, correct me if I’m wrong KFNot the same as the uncaused cause then? The first cause?

That’s a good question, but I would combine the two, an “uncaused first cause”

It is the first cause which in itself has no cause.

And it would be the progenitor of everything else

JVL

It is not a theological point. It is a matter of simple logic. I will give you a hint. Think of the difference between “necessary” and “contingent” beings. Then ask yourself this question: As a matter of logic, can all beings be contingent?

JVL,

there is a trilemma, tracing to Agrippa, who had warrant chains in mind: unspannable infinite regress vs question begging circularity vs a finitely remote unproved [proofs come after!] start point. A similar pattern holds for temporal causal order such as our cosmos (think for convenience years in succession). Where, obviously “were” there ever only utter non-being, as such has no causal power, that would forever obtain. So there is a world or better reality root . . . where it all comes from.

That brings us to infinite chain vs circular cause vs a finitely remote non-causal entity that is framework to any world existing.

Over a span of three years here, it was hammered out that the set we need to discuss this is the hyperreals mileposted by the hyperintegers: R* and Z*. Through the “catapult” 1/x function we can jump from transfinite hypereals such as H beyond any n in N we can count up to from 0 to h, a number closer to 0 than 1/n, n any counting number. To get negatives, we use additive inverses. That is, we can define an extended number line that is a unified line from tamed infinitesimals to transfinites. BTW, this opens up a different approach to Calculus, non-standard analysis. I have gone as far as to say, this is what our High School Math teachers were fishing for and it is what founders of the Calculus saw through a glass darkly. I think College Math should at least open the door a crack.

In that light you cannot count up from a transfinite negative hyperreal K’ s.t. K + K’ = 0 and attain to a finite range of 0:

. . . . K’ –> K’+1, K’+2 . . . K’+p –> K’+p+1, K’+p+2 . . . . -2, -1, 0, 1, 2, 3 . . . n, n being “now” and 0 conveniently the singularity. I use four dot ellipsis for a specifically transfinite span.

Notice, at K’+p, at some finite remove from K’, we are in effect starting the finite count over again, which can repeat any finite number of times. We cannot traverse a cumulative, transfinite span in finite stage steps.

So, the logic of structure and quantity plus the concept that any actual past stage had to have once been the actual present, leads to the problem. Namely, a supposed past point K’ cannot succeed in finite stage steps to 0 much less now, usually estimated at 13 – 14 BY beyond singularity. There was no transfinite remote actual past that was explicitly or implicitly transfinite.

World from nothing –> X

World from infinite past –> X

Circular cause is just as bad: the not yet reaches back and sets in motion the chain of stages that causally, temporally succeed to itself.

Circular cause –> X.

Last man standing: a finitely remote, world source or root that is not causally dependent. That is, a necessary being. We can call it an ultimate cause or THE first cause.

We need a necessary being reality root that is causally independent and adequate to account for any possible or actual world. Including a world with morally governed creatures — us.

From this, much follows.

KF

PS: Try to imagine a world where two-ness does not exist, or begins or comes to an end. That already tells you what a necessary as opposed to contingent being is like. And this is logic of being, ontology, a major branch of metaphysics. Philosophy, employing another major branch, logic. Not, theology. Where God enters is as the idea of God, the God of philosophy. Though, he’s familiar.

As78, “uncaused” is problematic, suggesting an arbitrary claim. Logic of being is a more basic approach as BA hints at. Start with a possible way this or any other world may be and turn it into a set of propositions that sufficiently describe. We can then consider the collective, possible worlds. An impossible being — due to contradictory core characteristics [think, square circle] — cannot exist in any PW. Possible beings would or do exist in at least one PW, where our actual world obviously is a PW. Of such, contingent ones exist in some but not all; we can add the difference between W with C and closely neighbouring W’ without C can be identified as an enabling causal factor for C. A necessary being, N, exists in any PW; best understood as, it is framework for any W to exist. Above, I gave 2 as an example. It is impossible to have any distinct possible world W without two-ness. So, we have in mind a necessary being reality root that has no enabling causal factors of its nature not by arbitrary stipulation. KF

@31

Thank you sir

AS, welcome. I add, note that a true nothing is non being. Were there ever utter non-being, there is no causal capacity and that utter non being would hold forever. A world now is, something adequate to account for a world always was. This is the root of reality, source of any actual world.Then, what is this leads to the causal chain trilemma discussed already. The only feasible root arises from the problems of infinite causal succession and circular causation. Finitely remote necessary being capable of being source for this or any other actual world. Where, we are morally governed creatures in an actual world, pointing to a root adequate to bridge IS and OUGHT. KF

KF

“that utter non being would hold forever”

Agreed, for the simple reason that non-being is obviously causally inert. “Nothing” cannot cause “anything.” This is one of those grounding plausibles of which you often speak. It is a sign of our times that some people do not accept it as necessarily true. All one has to do is give up on the law of sufficient reason, and poof “something from nothing.” So one can have his rationality or one can have his something from nothing . One cannot have both.

Barry Arrington: Then ask yourself this question: As a matter of logic, can all beings be contingent?IF I understand the question correctly I’d say yes, all living beings can be contingent. That’s based on my belief that life can come from non-life.

If I’ve misinterpreted the question tell me.

Kairosfocus: Where, obviously “were” there ever only utter non-being, as such has no causal power, that would forever obtain.I guess that’s a central point we disagree on since I believe life came from non-life. So there is no infinite regress to worry about!

KF

I’m aware that this is your belief.

Yes, humans have imposed a moral overlay on society. Sounds like a reasonable thing to do if you want to live in a stable society. It falls along a spectrum of behavioural expectations if you want to be part of society. At one end you have severe legal penalties for behaviours that most people agree seriously conflict with a healthy society (eg. Murder, violence, theft, etc. ). In the middle are minor departures from accepted behaviour that justify some structured penalty (eg. Fines for speeding, noise by-laws, refusing to bake a cake for a gay couple, etc. ). And at the other end are behaviours that can be classified as violations of trust but don’t warrant direct legal legal penalties (eg., infidelity, dishonesty, etc. ).

A root that is adequately filled by rational human beings.

But why impose an unproven god on something that human societies have, for centuries, decided for themselves?

See above.

Ed George, in response to,

,,states,,

So, let me get this right, apparently EG does not believe that “reality requires a necessary being world root” and therefore believes that all reality must be contingent???

But that can’t be right!

Maybe E.G, if he was not just being purposely unreasonable, just didn’t understand?

To put the question more simply, ‘What is the necessary being upon which all contingent being is dependent for its existence?’

To clarify,,

Further arguments:

JVL

Your statement reveals that you do not understand the terms used. Whether life came from non-life is completely irrelevant to whether, as a matter of logic, all beings can be contingent. It is as if I asked your what is the sum of 2+2 and you responded, “The Gettysburg Address.” While “The Gettysburg Address” may be the answer to some question, it has no bearing on the question I posed.

Barry Arrington: Your statement reveals that you do not understand the terms used. Whether life came from non-life is completely irrelevant to whether, as a matter of logic, all beings can be contingent.Well, what does contingent mean then in this context? I’ve looked up a general definition but I’d rather be sure I understand you correctly.

JVL, the issue is not all living beings [biological, cell based sense] but all beings. It is obvious that biological cells are constructed from atomic and molecular components. Such a composite entity is necessarily composite. Similarly, non-life (biological sense) in your context would be chemicals in some alleged prebiotic stew. A material substance like that is not non-being. Being is NOT, rpt, NOT, a synonym for biological entity. Going further, atoms are composite and contingent also. Cell based life is indeed contingent but it is by no means established in worldview analysis contexts that such exhaust the relevant sense of life. God, would be living, life himself indeed and ultimate author of biological life. Plato’s Athenian Stranger speaks of the ensouled, self-moved first cause agent as life. KF

EG, notice your reframing as personal belief. No, I provided an

analysison possible worlds and categorisation of beings across PWs. In that context of logic of being, I noted the causal futility of utter non being, where we live in a world so a world from non-being is a non-starter. I proceeded to the causal-temporal form Agrippa trilemma, eliminating infinite regress as non-traversible and circular causation (requiring the now to come from the not yet). The last man standing, per comparative difficulties, is a finitely remote, necessary being world root. That I accept the hard won conclusion is not mere personal credulity but warrant. That warrant is what you need to cogently address. KFPS: Your attempt to reduce moral government to a human epiphenomenon of the superstructure of society . . . the Marxism reeks . . . actually reduces to grand delusion and collapse of credibility of the mind. For example, even your own argument requires for any persuasive force, that we implicitly recognise that we are duty-bound to truth, to right reason, to prudence [including warrant!], to sound conscience, to fairness and justice etc. Were there mere epiphenomena, the import is Plato’s cave grand delusion manipulated by power brokers and their shadow show actors. The immediate collapse of responsible rationality and open invitation to nihilism are patent. Likewise, you have fallaciously substituted primacy of hyperskepticism for prudence, leading to characteristic modern and ultra-modern fallacies of selective hyperskepticism. You have again made shipwreck of your intellectual credibility.

PPS: That you wish to put humans at the reality root is a manifest failure of logic of being. Contingent beings that are far in the future cannot be the reality root. More intellectual credibility collapse. You seem to imagine that clever rhetorical stunts substitute for serious analysis. Fail.

.

Not to change the subject, but JVL, you obviously believe life began as a purely dynamic replicator (perhaps a “simple RNA replicator”) , and then evolved into the more complex semiotic replicator (modern DNA/RNA/aaRS-mediated replicator) that is described in biology today. Is that correct?

BA77, Good thoughts. It seems in various aspects EG is implying a world from nothing or else circular causation or infinite causal-temporal past without squarely facing the challenges such suggestions lead to. It is highly likely that he assumes we must be wrong so he is stringing together rhetorical stunts that snip bits and pieces out of context to dismiss what he has already determined to reject without serious consideration. His result is to shatter his own credibility. KF

JVL, maybe here will help: https://uncommondescent.com/mathematics/logic-first-principles-4-the-logic-of-being-causality-and-science/ KF

F/N: A helpful vid on trying to pull a cosmos out of a non-existent hat: https://uncommondescent.com/philosophy/l-fp41-dawkins-krauss-and-trying-to-pull-a-world-out-of-no-thing/ KF

KF

Exactly. Seversky earlier today brought up the issue of fideism. Well, fideism is alive and well among don’t-look-in-that-direction materialists such as EG. The glaringly obvious hole in his worldview is there for all to see. Yet, he accuses theists of grit-your-teeth blind faith. Irony alert.

KF

Not reframing, just stating a fact.

Nice verbal masturbation, but all you have said is that, because we don’t know the cause of the universe (or life) that, therefore, god is the best explanation. There is a term for that.

Again, there is a term for your logic.

I guess that I have a higher opinion of the abilities of humans than you do.

No, it requires that we have the ability to reason, without resorting to a mythical being, that it is in our best interest to be truthful and to treat others as we would like them to treat us. I know that I can reason this from first principles. I’m surprised that you feel that you can’t.

Translation: you disagree with me.

Thank you for misrepresenting what I have said.

When you start presenting serious analysis, please be sure to let me know.

EG, KF, if I may jump in:

Could it require nothing more than humans acting in their own long-term best interests? Where we are informed by our own experience and the writings of those who have reflected on how best to live over the past few thousand years?

EG, we know the rhetorical stunt that you used as a wedge to evade grappling with the analysis then chop up into bits and pieces then play strawman games. Your onward remarks, including vulgar insulting references and dismissals simply underscores the self-destruction of your intellectual credibility. Repeat, we live in a causal temporal world, one that includes inescapably morally governed creatures. Infinite regress of causal stages cannot be successively traversed, worlds do not come from nothing, and circular cause is another form, trying to pull the past of origins out of that past’s not yet successors. The last man standing is a finitely remote necessary being reality root, sufficient to account for this cosmos, including morally governed creatures. That you are hyperskeptically dismissive of such is a psychological and ideological fact about you and your ilk but it has nothing to do with the actual balance on merits. Where, the no evidence or no analysis selectively hyperskeptical gambit inadvertently reveals the fallacy of selective hyperskepticism and your want of a cogent response. Duly noted, FAIL again. KF

DS, no. Our best interests is already an appeal to such first duties. What is inescapable is inescapably true as Epictetus pointed out 1850 years ago to one who challenged him to prove the validity of logic. KF

PS: Epictetus:

Barry,

“Exactly. Seversky earlier today brought up the issue of fideism. Well, fideism is alive and well among don’t-look-in-that-direction materialists such as EG. “

So true and fun to watch the science deniers and the fideists accuse the non science deniers and the non fideists of being that which they are and that which we are not.

As to logic it is a sight to behold the ignorance on display here by EG and others.

Vivid

KF,

I’m not sure what this means exactly, but let’s pursue this. Are we unable to discern these “first duties” without the existence of a god?

JVL

“I guess that’s a central point we disagree on since I believe life came from non-life.”

Yeh and and hundreds of years ago people thought maggots came from meat.

Vivid

DS, back ways around. We find ourselves situated much like Epictetus’ interlocutor. We find that some things are inescapable so true antecedent to proofs. They are foundational to proof making etc. Then, we can ask, what sort of world, from the roots, sustains that? That sets up a bill of requisites for the world root. KF

KF,

Hm. I guess I don’t feel the need to posit the existence of this world root.

Everything that would require sustenance is apparently sustained by what I can already observe, my relationships with others, for the most part.

DS, it is not a positing, it is an analysis across three often unpalatable options. That the world is causal-temporal and successive is a manifest commonplace. So, how do we trace that back? Infinite traverse from a transfinite past is one candidate, failed. Circular cause is another, also failed. A world from utter non-being is a non-starter. That leaves a finitely remote world root as the last man standing. Where, your own onward arguments implicitly turn on the same first duties of reason as identified. The issue may not be one one wishes to explore, but if one is interested seriously in origins of reality and how things came to be, such things are there. KF

Vividbleau@ 17Probably not. Christianity embraces a range of different denominations and doctrines, not all of which are consistent with one another. Unfortunately, over the centuries, these differences have led to a lot of Christian blood being spilled by other Christians who believed they were doing the Lord’s work. That behavior may have found warrant in the morality of the Old Testament but I do not believe it is what Christ intended or taught. For example, the intolerance I see displayed here towards theistic evolutionists is distinctly un-Christian.

KF,

I don’t even know what a world from utter non-being would mean, so I agree that option should be dismissed. Whether your list of options is exhaustive is questionable.

I’d rather not get into another another discussion of these issues if at all possible.

I take it you identify fairness as one of these “first duties”?

What role does this world-root being have in establishing this duty? How exactly does the “sustaining” occur? As far as I can tell, no one needs to tell me I should be fair to others (at this point in my life, anyway). Rather, that’s the conclusion that we as humans have come to, through thousands of years of experience. Furthermore, whether I treat others fairly or not actually does have an impact on the quality of my life, so I have a reason to do so.

DS, propose another. Skeptical dismissal does not hold default. KF

KF,

For the sake of my sanity, I’m going to decline (only half-joking 😛 )

I withdraw my questions.

KF

Agreed.

That is your belief, not fact. I agree that humans have a moral sense. But to say that we are inescapably morally governed is just wishful thinking. My moral values are different than yours. And mine have changed over the years, as I assume yours have as well. And I think that it is important that they are plastic and change as society changes. If not we would still think that segregation is morally acceptable, or that it is morally acceptable to provide women less opportunity than men, or that it is morally acceptable to deny same sex couples from marrying. All morals that have changed within my life time.

Ba77@10 wrote:

2. If it is possible that a maximally great being exists, then a maximally great being exists in some possible world.

3. If a maximally great being exists in some possible world, then it exists in every possible world.

Why does 3 follow from 2 ?

Kairosfocus: the issue is not all living beings [biological, cell based sense] but all beings. It is obvious that biological cells are constructed from atomic and molecular components. Such a composite entity is necessarily composite. Similarly, non-life (biological sense) in your context would be chemicals in some alleged prebiotic stew. A material substance like that is not non-being. Being is NOT, rpt, NOT, a synonym for biological entity. Going further, atoms are composite and contingent also. Cell based life is indeed contingent but it is by no means established in worldview analysis contexts that such exhaust the relevant sense of life. God, would be living, life himself indeed and ultimate author of biological life. Plato’s Athenian Stranger speaks of the ensouled, self-moved first cause agent as life.Well, I don’t know how my views fit into your questions. Like I said: I believe that life arose from non-living substances without any guidance. Whether that is contingent or not . . .

Upright BiPed: Not to change the subject, but JVL, you obviously believe life began as a purely dynamic replicator (perhaps a “simple RNA replicator”) , and then evolved into the more complex semiotic replicator (modern DNA/RNA/aaRS-mediated replicator) that is described in biology today. Is that correct?I’m still not sure if the term “semiotic” matches my thinking but . . . yeah, basically.

Kairosfocus: JVL, maybe here will help: https://uncommondescent.com/mathematics/logic-first-principles-4-the-logic-of-being-causality-and-science/I guess then, in my view, human beings are contingent . . . on the right environmental conditions.

Vividbleau: Yeh and and hundreds of years ago people thought maggots came from meat.It’s a slightly different view! They thought maggots rose directly from meat not through a millions of year process. And you can’t blame them really. If you’ve ever left a piece of perfectly good looking meat out it does seem like the maggots come from nowhere else.

Kairosfocus: Circular cause is another, also failed.Does that mean you reject the idea that the universe MAY expand, stop, contract, go through another big bang, expand, etc?

.

The term semiotic merely refers to a description-based replication versus the RNA dynamic-based replication. There is nothing controversial about that distinction, given that the constituents of each system are clearly known, and their roles are understood.

So assuming a continuum of primary function (self-replication), you then believe that there was once a point in time that a chemical organization existed on earth that could successfully replicate itself both dynamically and as a description-based replicator at the same time. Is that correct?

EG, personalisation again, while refusing to look at the readily seen point that your own attempted counter argument itself relies implicitly on our known duties to truth, right reason, prudence [so, warrant], sound conscience, neighbour [so, fairness and justice] etc. Kindly look at Epictetus again. The inescapable is inescapably true. KF

ES58, BA77 is citing someone, Kreeft. The suppressed point is that a maximally great being is a serious candidate necessary being. As such, it would either be impossible of being or actual and if the latter framework to any possible world. In that context, possible means, in at least one world were it actual. From that, necessary and present in every possible world including such as are actualised. See discussion here, no 4 in the L&FP series: https://uncommondescent.com/mathematics/logic-first-principles-4-the-logic-of-being-causality-and-science/ KF

JVL, biological life is based on molecular and atomic components so is necessarily contingent. Contingent, implying causally dependent on antecedents, being within a world and at least potentially subject to destruction by disintegration of sufficiently critical parts. The OOL debate I defer to another time. Being here goes beyond biology. KF

JVL, I happen to note the observed flatness and that suggests further that oscillating cosmos is off the table. Besides, accumulating entropy would limit its number of cycles, up to 100, each bigger than the last is what I have seen. That is not the sense of circular cause in view. The idea in view is that a yet to be future causal-temporal state somehow reaches back — not merely, logically constrains — and triggers origins. This is a simple logical option for addressing causal-temporal succession of stages. KF

UB, RNA world or the like of course run into the challenge, how conveniently do we get complex machine code and execution units out of molecular chaos. Especially when the processes are thermodynamically unfavourable. Have they pondered ATP as energy currency of the cell? KF

Es58 at 63:

Dr Craig answers that question here:

Also see the argument for God from contingency and time in post 38

https://uncommondescent.com/philosophy/godels-proof-of-the-existence-of-god/#comment-701484

The beauty of this argument, as Dr Craig goes on to note here, it that it “puts the atheist in a very awkward position. The atheist must deny, not merely that God exists, he must maintain that it is impossible that God exists. And that is certainly a radical claim that would require great justification.”

And as pointed out in post 10, where this gets VERY interesting is that atheists, in their appeal to an infinity of other possible worlds to try to ‘explain away’ the fine tuning of this universe,,,

,,, in their appeal to a multiverse to try to ‘explain away’ the fine tuning of this universe, atheists have, basically, completely conceded the necessary premise of step 1, i.e. that it is possible that God exists, to the ontological argument in order for the ontological argument to work.

Simply put, you cannot argue that it is possible that an infinity of other universes exist in which an infinity of other possibilities are playing out, while at the same time holding that it is impossible for a maximally great being to exist in one of those infinity of other universes:

The absurdity of the atheist’s predicament with his appeal to the multiverse is humorously laid out in the following article,

Folks, I must note the on the whole serious and substantial tone of discussion. KF

BA77, a key problem is that logic of being and possible worlds are not commonly taught today, so are unfamiliar. Similarly for logic. We have been so steeped in scientism that we do not realise things that are antecedent to science itself, or to Math, or to reasoning. KF

PS: SEP on PW https://plato.stanford.edu/entries/possible-worlds/ Also, Wikipedia has a helpful summary https://en.wikipedia.org/wiki/Possible_world Try this too http://www.cs.ox.ac.uk/people/.....3_2_3.html

Kairosfocus: biological life is based on molecular and atomic components so is necessarily contingent. Contingent, implying causally dependent on antecedents, being within a world and at least potentially subject to destruction by disintegration of sufficiently critical parts.Are we disagreeing then? I guess it depends on what parts you’re thinking of. You could be thinking of a fine tuning tact which is not what I thought we were discussing.

I happen to note the observed flatness and that suggests further that oscillating cosmos is off the table.Flatness? I thought the universe was still expanding? Anyway, new data may come along so I suppose we can’t really be sure just yet.

Besides, accumulating entropy would limit its number of cycles, up to 100, each bigger than the last is what I have seen.Really? Where did you see that? I haven’t come across that notion before.

That is not the sense of circular cause in view. The idea in view is that a yet to be future causal-temporal state somehow reaches back — not merely, logically constrains — and triggers origins. This is a simple logical option for addressing causal-temporal succession of stages.That’s not what I was thinking you meant . . . I’m not sure what it means actually.

Folks, I must note the on the whole serious and substantial tone of discussion.The way it should be!

JVL,

The point at stake is not origin of biological, cell-based life. I happen to think the common speculations are ill-informed, skip over serious issues of origin of functionally specific complex organisation etc but such are secondary. Bio life is inherently composite and so is by virtue of that inherently contingent. Necessary being is not like that.

The flatness means, not enough mass etc to contract. Similarly, I am talking of stuff from 20 years back, that is actually glorified common sense. An oscillating cosmos will be subject all along to dissipation of concentrations of energy. Less around to drive the cycle, takes more time to come back together, eventually, runaway of cycle length. In any case, on logic of structure and quantity, an eternal past physical causal-temporal world cannot escape traversal of the transfinite.

Circular cause is causal question begging.

KF

Kairosfocus: The point at stake is not origin of biological, cell-based life. I happen to think the common speculations are ill-informed, skip over serious issues of origin of functionally specific complex organisation etc but such are secondary. Bio life is inherently composite and so is by virtue of that inherently contingent. Necessary being is not like that.Okay . . .

The flatness means, not enough mass etc to contract. Similarly, I am talking of stuff from 20 years back, that is actually glorified common sense. An oscillating cosmos will be subject all along to dissipation of concentrations of energy. Less around to drive the cycle, takes more time to come back together, eventually, runaway of cycle length.Where would the energy go? IF the universe is oscillating then surely nothing is leaking out of the system so . . .

In any case, on logic of structure and quantity, an eternal past physical causal-temporal world cannot escape traversal of the transfinite.I’m not sure I agree on that but I have looked over your points so please don’t feel the need to repeat them.

Circular cause is causal question begging.I’m not sure where to go then. I don’t see the need for a necessary being (which presumably kick-started everything else which then could be contingent). I’ve read your points and I don’t find all of the conclusions solid.

I’m not saying the universe came out of nothing but I don’t see why it couldn’t have arisen through some kind of natural, unguided process.

And, to tell you the truth, I’ve never really got ‘where’ some necessary being that lives outside of space and time would exist. In order for such a being to influence our physical realm then it must partly exist in our physical realm. Where? Where does its energy come from? How can it affect physical reality? I don’t see how all that would work.

JVL, energy does not vanish, it gradually degrades through loss of concentration; the heat death idea illustrates. On past infinity, our opinions are irrelevant, the force of structure and quantity is decisive — we cannot traverse a transfinite in finite stage steps, period. I next suggest, that you ponder why a square circle is impossible of being, and also a fire as contingent on heat, oxidiser, fuel and combustion chain reaction. then consider something like the number 2, which reflects distinction A vs NOT-A and so so soon as a distinct thing exists, duality necessarily obtains. The concepts of impossible, contingent and necessary being are not arbitrary opinions. As for a spatial-temporal vision of reality, it is not that necessary beings are on the outside but that they are the framework for a world such as ours to exist. A problem is that as a dominant ideology teaches, we have been led to imagine that the space-time world we inhabit is the limit of reality and being. So yes you are struggling with a paradigm shift, including, to what extent do you trust logic, including logic of structure and quantity, aka mathematics? KF

Vivid,

You put on the table, logic.

By implication, first principles of right reason, especially the core, distinct identity.

There is a distinct thing A, it is “in” reality. we recognise it by marking that it is distinct from what it is not, ~A.

We partition reality, conceptually: R = {A|~A}. From that, immediately, A is itself i/l/o core characteristics. No particular x in R is both A AND ~ A in the same sense and circumstances. Any particular y MUST be A or ~A, not in the middle or neither. LOI, LNC, LEM as corollaries.

Also, the partition has two distinct units, A and ~A, A simple unity, ~A complex. The partition | is empty. Thus, we see nullity, duality, unity, and can apply von Neumann:

{} –> 0, the set that collects nothing is a distinct and unique entity, the null set, defining cardinality 0.

{0} –> 1, the set that collects the null set is distinct from nullity and has cardinality 1, it is ordinally the first number beyond 0

{0,1} –> 2, we have two distinct units and can collect so, 2 similarly, second natural counting number.

{0,1,2} –> 3, the succession builds up and chains without limit

. . . , chaining, and so infinity appears

{0,1,2,3 . . . } –> w, omega is order type of the naturals and our first recognised transfinite.

Logic has laid out a world of abstracta that obtain in any possible world, they are framework for any world. These are necessary beings.

Much more can be elaborated.

KF

KF

Our duties of truth, right reason , prudence, etc. are self imposed. Again, I can reason from first principles that the best strategy if I want to enjoy the benefits of living in a society is to generally be truthful and treat others as I would like to be treated (i.e., fairness). Those are things I choose to do, not something that I am compelled to do, other than as the result of societal and legal pressures.

KF,

What is your response to the claim that the so-called necessary being we call “twoness” does not actually exist outside our minds? Rather that it’s a mental construct that (at least some) intelligent beings use to navigate the world? (I’m implicitly using a similar construct when I say “at least some” of course, but I don’t know how to avoid this. Or even if it’s desirable to avoid it).

My concern is that it just seems too easy to reel off a bunch of (oops, did it again) abstract concepts and claim they are necessary beings. And further, very little is required of the claimant. How do you defend the claim that a particular abstract entity really exists outside our minds? “If you don’t get it, you’re deluded” is about the strongest argument I can think of. (Note: I’m happy to talk about math-related things, but don’t want to get embroiled in the timing/existence of the origin of the universe right now).

EG, our duties are mutually recognised. If self imposed, optional and ending in nihilistic chaos. Which is exactly what Plato warned about in The Laws and in Republic VI, ie ideologies that create that impression are ruinous. Indeed, if self imposed, little more than a confession of powerlessness so solution is grab and impose will by power; nihilism. Further, if self imposed and believed, delusional and undermining credibility of mind. Further, notice how in arguing as you do you implicitly appeal to our sense of being under those first duties, i.e. inescapable. Fail, yet again. KF

Kairosfocus: energy does not vanish, it gradually degrades through loss of concentration; the heat death idea illustrates.I’d say it dissipates rather than it degrades but it sounds like we’re thinking the same thing. But I still don’t see how a closed system universe that is oscillating would gradually have less energy. A spring or pendulum eventually slows down because it is losing heat to the environment but the universe has no environment to lose energy in to; it is a closed system. I’m not saying the universe is oscillating but if it were (starting with a singularity, expanding and then collapsing again into a singularity) then all the energy in the system stays in the system and gets put back into the singularity.

On past infinity, our opinions are irrelevant, the force of structure and quantity is decisive — we cannot traverse a transfinite in finite stage steps, period.Again, if the universe is oscillating then why would there be an end in either direction? You never get to the transfinite but it doesn’t mean there’s a end or a detectable beginning.

I next suggest, that you ponder why a square circle is impossible of beingThat’s just a matter of definition isn’t it? A square is defined in a certain way and a circle is defined in a different way.

and also a fire as contingent on heat, oxidiser, fuel and combustion chain reaction. then consider something like the number 2, which reflects distinction A vs NOT-A and so so soon as a distinct thing exists,Umm . . . I don’t see what 2 has to do with A vs not-A since 2 is just a distinct number, certainly not the same as any of the other infinite number of distinct numbers.

The concepts of impossible, contingent and necessary being are not arbitrary opinions.Personally, I can’t see that a necessary being is necessary. Sorry, I know you’re trying to explain but I can’t see a hard and physical reason why it should be so.

As for a spatial-temporal vision of reality, it is not that necessary beings are on the outside but that they are the framework for a world such as ours to exist.Are you saying the universe is, in some sense, the necessary being or part of the necessary being? If not then the necessary being exists outside the universe? But it interacts with the universe? It expends energy in the universe? Does it add energy to the universe or just move it about? If it adds energy where does that energy come from? If it’s moving energy about then how does it do that without some physical manifestation or mechanism? And if there were a physical manifestation or mechanism then we should be able to perceive it I would think.

A problem is that as a dominant ideology teaches, we have been led to imagine that the space-time world we inhabit is the limit of reality and being. So yes you are struggling with a paradigm shift, including, to what extent do you trust logic, including logic of structure and quantity, aka mathematics?I don’t see that you’ve offered any strong and compelling reasons to think there is anything outside of the known universe though! Logic is a human construction and so is fallible when applied outside of its realm. Mathematics I do trust but you haven’t done much of that. You can talk about transfinite or infinite cardinal numbers and I agree you can’t get there in a finite number of steps (that is the point after all!) but I don’t see how that kind of mathematics applies to a potentially oscillating universe, each oscillation which would have a life-span and would not interact with or affect other oscillations. A perpetual motion machine as it were. Again, I’m not saying that is the case but I do consider it a possibility.

Sorry if I’ve misrepresented anything you’ve said or if I’ve not grasped a concept that seems crystal clear to you. I am trying to follow your arguments but something is just not resonating in my head.

DS, is there a difference between an egg and a stone or a poisonous scorpion? Distinction, distinct identity are not mental pictures projected to the world. Nor is it an arbitrary list of claims. One has to warrant serious candidacy. Structure and quantity including nullity, simple and complex unity, duality etc are part of the rationally intelligible framework of reality; it is that we are language using intelligences able to grasp more than other creatures such as birds or monkeys etc. As for root thereof, we face the causal-temporal Agrippa trilemma, and that sharply constrains options. KF

JVL, I have to move on. I actually thought on that word choice and concluded degrade is more accessible. I of course allude to entropy and its statistical underpinnings. Gotta go now. DV, later. KF

DaveS: What is your response to the claim that the so-called necessary being we call “twoness” does not actually exist outside our minds? Rather that it’s a mental construct that (at least some) intelligent beings use to navigate the world?I’ve heard some mathematicians discussing such issues but usually only after a few beers. I’ve come to the conclusion there is no answer to that although personally I think that everywhere in the universe the mathematics has to be the same (discuss!) and so it exists outside ourselves. The order we discover mathematical truths has been a function of culture until fairly recently, i.e. different cultures worked on different aspects of mathematics depending on what problems they were trying to solve. Gamblers developed some of the basic laws of probabilities for example. The cool thing about mathematics is that anything that was true in the past is still true today and so once the core mathematical results were disseminated over the whole planet then mathematical research started to break free from cultural influences.

Kairosfocus: I have to move on. I actually thought on that word choice and concluded degrade is more accessible.Yeah, I got that.

I of course allude to entropy and its statistical underpinnings.Yup, I got that too!

Gotta go now. DV, later.Okay! (I wonder what DV means . . . delay something . . . discuss something . . . could it be from Latin? )

KF,

Remember, you’re talking to a human here, who habitually uses numbers to count things, and who is not (presently) some sort of monist .

But if showing that abstract objects are necessary beings is as simple as in your post #82, I’m happy to go to town. (Well, in truth, it suggest to me these claims are perhaps empty).

PS: Let me add, through dissipation it becomes degraded.

DS, I am heading out the door now. NBs are framework to possible worlds, which makes them anything but trivial. The framework character of structure and quantity is what gives Mathematics its analytical and predictive power without its exerting active causal powers. That is an answer to Wigner, which is no trivial issue. It exercised my mind for years. KF

JVL,

I think that’s a reasonable argument, and is one I would cite in defense of some variety of neoplatonism. I’m becoming more sympathetic to the view that there is “no answer” (accessible to humans, anyway).

No response to #69?

Before I forget I’m just going to toss out one more thought . . .

I think the concept of distinctness works very well for abstract things like ‘2’ but I don’t think it works well for many real world examples.

Consider a chair. We can all probably agree on which objects are chairs and which are not (although I find stools and ergonomic chairs kind-of pushing against the boundary). Maybe with some extra linguistic specification we can tighten up the boundaries a bit. But what about when we look at a particular chair. Can you clearly delineate a particular chair from another, like if you bought a set of them. Probably it’s easiest to differentiate them based on their cosmetic differences or ‘flaws’; that’s the one with a scratch on the back, that’s the one with a stain on the seat. But what if we had two freshly made wooden chairs and there were no visible distinguishing marks? Okay, we could look at the particular molecular or atomic arrangements, they would not be the same. But once we’re down to that level then a chair begins to look a lot like part of a tree: same molecules mostly arranged in the same way. In fact on the molecular level how can you distinguish between an all wooden door and an all wooden chair? If I take a door and cut it up and make a chair out of it the atoms and molecules are exactly the same but now I have a new thing, with a new identity? So arrangement is key.

If Captain Kirk got into the transporter and the damn thing malfunctioned, again, and he arrived at his destination just a pile of molecules then we wouldn’t say it’s Captain Kirk anymore even though all the parts are the same. Or are they . . . if the transporter dissembles Captain Kirk and then puts him back together again on the atomic level are even the molecules the same? Even worse, if the ‘new’ Captain Kirk is just an atom-for-atom copy of the old Captain Kirk (who has presumably been destroyed) then he might walk like Kirk, talk like Kirk and act like Kirk but is he the same thing? On the other hand if we saw Captain Kirk once and then again 10 years later we would say he is the same man even though most of his atoms and molecules had probably been replaced during that time.

That was a bit rambling but I hope I got my idea across: differentiating between things is, at some level, hard or arbitrary. Perhaps in the Platonic mind-world there is a perfect square or rabbit or Captain Kirk. But in the real world it’s all a bit messy. It’s not just parts, it’s not just the arrangement of the parts.

Upright BiPed: No response to #69?I’m so sorry, I missed that comment!

The term semiotic merely refers to a description-based replication versus the RNA dynamic-based replication.So, abstract? Not decided physically?

There is nothing controversial about that distinction, given that the constituents of each system are clearly known, and their roles are understood.So, no chemical affinities at all?

So assuming a continuum of primary function (self-replication), you then believe that there was once a point in time that a chemical organization existed on earth that could successfully replicate itself both dynamically and as a description-based replicator at the same time. Is that correct?If the description was not purely arbitrary but was developed via some basic chemical influences . . .

I guess I just see the whole thing coming down to physics and chemistry; binding sites, particular sequences and environmental influences and resources. Which would mean I’m questioning the use of semiotic to describe the system. When I looked up semiotic I found this:

That seems to imply a purely abstract or arbitrary assignment of meaning.

IF the genetic replication system is semiotic and there are no chemical reasons for it being what it is (so it could have been something else entirely) then . . .

Could it not still be the product of purely physical processes? A frozen moment that worked?

BUT . . . if it’s purely arbitrary with no chemical properties then . . how does it work? I mean it translates via binding sites and a particular sequence which are physical processes. And if you change a DNA sequence you can get a different output so the physical configuration matters.

IF the genetic process was arbitrary then why couldn’t it just change willy nilly? Would we see lots and lots of different genetic codes? (Assuming they’re unguided.) And if they are guided then does that mean there is some guiding mechanism each and every time DNA or RNA do their thing to make sure the purely arbitrary code is followed?

Perhaps I don’t fully understand how it all works. I’m no geneticist that’s for sure. Please don’t take offence if I’ve mucked it all up; I am trying to understand.

DS & JVL, actually, such questions are addressed and they have significant force. there is a reason why so many Mathematicians and the like are Mathematical Platonists. Here is my own comment on the matter: https://journals.blythinstitute.org/ojs/index.php/cbi/article/view/62/59 KF

PS: Borrowing some WiFi

KF,

That's true. Perhaps this is evidence that some form of mathematical platonism is true.

On the other hand, there are also reasons why so many popes have been Catholic.

DS, there is a definitional reason why Popes are Catholic; that is not the case with Mathematicians. KF

.

#97

The question I am asking does not turn on what mechanism established the modern system.

Allow me to restate the question in a way that removes your aversion to the word “semiotic”. I’ll make the distinction (as I thought I already had) between the types of systems themselves (the speculated

“simple RNA replicator”and“modern DNA/RNA/aaRS-mediated replicator”) — that is, a distinction between the system as it is actually found today and the system which is speculated to have been the original (regardless of whether of modern system came about by frozen accident, or chemical affinity, or any other mechanism). Surely you grasp that the system would have to change from one system to the other if the system that is found in the cell today is deemed to be too complex to have arisen spontaneously, so another simpler system is speculated to have been the original replicator. And it also seems clear that even though proponents (who speculate a simpler primitive system) have not been able to put forward an experimental or theoretical demonstration of how the change was to have occurred, they must still expect, understand, and concur that the system would indeed have to change from the primitive to the complex – which is the whole point of their speculation.Having said all of that, I must say I find it wasteful to have to dance around these terms (K. Popper) when it is abundantly clear that we both know what we are talking about. It reminds me of the striking difference I found when I first started researching these topics on my own. When I read a paper that I enjoyed or challenged my understanding, I would not hesitate to contact the author directly and kick off a conversation. I found it striking that these kinds of word-derby situations never took place. In any case, here is the restated question below. If you again choose not to answer (given the obvious nature of the question being asked) I will assume this is yet another among the many key topics and observations entailed by your position that (as you’ve already said) you are not interested in addressing. I will abandon the question (because I am fairly certain that KF would rather I not clog up his combox with this topic anyway)..

Assuming a continuum of primary function (self-replication), you then believe that there was once a point in time that a chemical organization existed on earth that could successfully replicate itself both as a non-DNA/RNA/aaRS-mediated replicator and as a DNA/RNA/aaRS-mediated replicator? Is that correct?

KF,

Yes, but there are parallels as well.

Upright BiPed: The question I am asking does not turn on what mechanism established the modern system:Okay.

Allow me to restate the question in a way that removes your aversion to the word “semiotic”. I’ll make the distinction (as I thought I already had) between the types of systems themselves (the speculated “simple RNA replicator” and “modern DNA/RNA/aaRS-mediated replicator”) — that is, a distinction between the system as it is actually found today and the system which is speculated to have been the original (regardless of whether of modern system came about by frozen accident, or chemical affinity, or any other mechanism).Go on . . .

Surely you grasp that the system would have to change from one system to the other if the system that is found in the cell today is deemed to be too complex to have arisen spontaneously, so another simpler system is speculated to have been the original replicator.Why should I grasp that? Who has deemed it to have been too complex? Now you’re delving into opinion and not science. Maybe instead of dancing around the issue you would prefer to ask me outright.

And it also seems clear that even though proponents (who speculate a simpler primitive system) have not been able to put forward an experimental or theoretical demonstration of how the change was to have occurred, they must still expect, understand, and concur that the system would indeed have to change from the primitive to the complex – which is the whole point of their speculation.

I can’t speak for them but I can ask you to tell me what your alternative hypothesis is so that I can consider both proposals.

Having said all of that, I must say I find it wasteful to have to dance around these terms (K. Popper) when it is abundantly clear that we both know what we are talking about.There’s no need to dance at all; state your case clearly.

If you again choose not to answer (given the obvious nature of the question being asked) I will assume this is yet another among the many key topics and observations entailed by your position that (as you’ve already said) you are not interested in addressing. I will abandon the question (because I am fairly certain that KF would rather I not clog up his combox with this topic anyway)..Good.

Assuming a continuum of primary function (self-replication), you then believe that there was once a point in time that a chemical organization existed on earth that could successfully replicate itself both as a non-DNA/RNA/aaRS-mediated replicator and as a DNA/RNA/aaRS-mediated replicator? Is that correct?Explain to me how such an organism could be non-DNA/RNA mediated? How would that work? How would the mediation take place? What would be the mechanism?

KF

JVL, 86:

>>if the universe is oscillating then why would there be an end in either direction?>>

— A no beginning causal-temporal system would be infinite in the past. The ACTUAL past.

— That is why I point out that actual past requires that every past stage was once the present but now has been superseded by subsequent ones, cumulatively to now.

— An actual past stage infinitely remote from now could never have been succeeded by a transfinite number of steps

— Such a succession for all we know is without limit future-wise, it is POTENTIALLY infinite if so, but at any given now there cannot have been transfinitely remote past stages.

— In short, a finitely remote beginning is baked into an actual past that was once the now, finite stage causal-temporal world. (This was further discussed above and in linked.)

>>You never get to the transfinite but it doesn’t mean there’s a end or a detectable beginning.>>

— What is logically necessary on true premises is true. Our ability to detect the beginning is irrelevant.

–> The issue is not to get to the transfinite but to arrive here FROM a transfinitely remote actual past.

>>That’s [impossibility of a square circle] just a matter of definition isn’t it? A square is defined in a certain way and a circle is defined in a different way.>>

— The definition reflects key characteristics giving squares and circles their distinct identity, these definitions are non-arbitrary and we cannot toss core characteristics at whim

— No single entity can be both a square and a circle as the core characteristics are mutually contradictory.

>>I don’t see what 2 has to do with A vs not-A since 2 is just a distinct number, certainly not the same as any of the other infinite number of distinct numbers.>>

— As pointed out, duality is about a distinct character.

— Reality R has some distinct thing, A. What is not A is then ~A. A is a simple unit, ~A is a complex one, different from A. Consider together and duality drops out.

— Two-ness and so the number are framework characteristics for any possible world.

— That’s a non trivial result and goes to what number is.

— Where this leads to is the unreasonable effectiveness of Math in the world and in the Sciences especially. See discussion as already linked: https://journals.blythinstitute.org/ojs/index.php/cbi/article/view/62/59

>>I can’t see that a necessary being is necessary . . . I can’t see a hard and physical reason why it should be so.>>

— These are unusual topics.

— A necessary being will not have a PHYSICAL reason for that necessity as such are antecedent to actual physical worlds. Their reasons have to do with the logic of any world being possible or actual, that is the reasons lie in logic of being.

— A necessary being will be part of the framework for any world that may exist or exists. We just saw it for numbers and it extends through N, Z, Q, R and C too, defining a structure for quantity in any possible world. That gives Math conclusions their powerful generality. By the logic of what being a world is, such must obtain. Math (more particularly a core tied to the Euler identity 0 = 1 + e^i*pi) is as valid in Flatland or Narnia as in our world.

— that’s an astonishing result and it gives great power to Physics, Chem, Computing etc

>>Are you saying the universe is, in some sense, the necessary being or part of the necessary being?>>

— Just the opposite! As other possible worlds are reasonably conceivable and i/l/o a key marker, the singularity etc, our observed cosmos is credibly understood as contingent/

>> If not then the necessary being exists outside the universe?>>

— there is no one necessary being — infinitely many numbers as an example — but collectively they define the framework IN which we live, move and have our being

>> But it interacts with the universe?>>

— Numbers, for one, reflect logic of being constraints on this or any other universe. That’s not an interaction, it is a constraint tied to existence itself.

— God, a serious candidate NB, would be causal: creating and sustaining the world we live in.

>> It expends energy in the universe?>>

— Numbers constrain what is logically possible, they don’t have to be active agents. Similarly, logical corollaries of distinct identity mean that any distinct A cannot in the same sense and circumstances be also ~A.

— God would be author of the world and creator of its energy. That the creator-sustainer would work into and interact with his creation should not be astonishing.

— and more

>> Does it add energy to the universe or just move it about?>>

— No one NB of course. Math and similar logic of being frameworks and constraints don’t need energy to have effect. What is impossible logically cannot be, e.g. a square circle.

— God, presumably can add and move around energy. That is beyond the scope of thermodynamics as a going concern world discipline.

>> If it adds energy where does that energy come from?>>

— Logic of being realities don’t work by energy. The creator is author of energy as Creator.

>> If it’s moving energy about then how does it do that without some physical manifestation or mechanism?>>

— Logic of being is not using energy. God as creator put energy where it was in the first place, regulates how it moves about as sustainer and author of laws of nature. If he acts in unusual ways, there is no reason why that should be in principle undetectable.

— BA77 loves to talk of a candidate case, the radiation that burned an image into the surface of a certain shroud that just may be authentic i/l/o the match with the Sudarion of Oviedo.

>> And if there were a physical manifestation or mechanism then we should be able to perceive it I would think.>>

— I would think so, in at least some cases.

>>I don’t see that you’ve offered any strong and compelling reasons to think there is anything outside of the known universe though! >>

— H’mm, there is considerable discussion of a multiverse and what I have suggested is in the first instance open to that. Most obviously in setting the singularity at 0 on a causal temporal succession sequence

— There MAY be an at least quasi-physical cosmos beyond our observable gamut. I am open to such.

— The conceptual problem is your reversal: the observed cosmos is within the framework of reality and its structuring of possible worlds.

>> Logic is a human construction and so is fallible when applied outside of its realm. Mathematics I do trust but you haven’t done much of that.>>

— Math is best understood as logic of structure and quantity i.e. as subject matter, as discipline it is a study of same, including when axiomatic systems are used to spin out abstract possible worlds.

— Logic in the core we are discussing is NOT an arbitrary mind game, though that impression is often given. For a distinct A to be in reality, it has core characteristics that mark it apart from ~A. LOI, LNC, LEM are instant, mutually supporting corollaries from that.

— Such logic and its consequences is tied into the framework of reality. It is discovered/acknowledged by us, not our invention.

>>You can talk about transfinite or infinite cardinal numbers and I agree you can’t get there in a finite number of steps (that is the point after all!)>>

— a point of agreement.

>>but I don’t see how that kind of mathematics applies to a potentially oscillating universe, each oscillation which would have a life-span and would not interact with or affect other oscillations. >>

— that is just what would NOT be the case. An oscillating cosmos would be part of a connected, cumulatively acting causal temporal system and energy and energy effects from cycle n would affect circumstances in cycle n+1 etc.

>> A perpetual motion machine as it were. Again, I’m not saying that is the case but I do consider it a possibility.>>

— That is, you consider a different world order in which, say, statistics of atoms, molecules and particles . . . the foundations of thermodynamics . . . would be drastically different. (I don’t know how you would prevent sensitive dependence on initial conditions and other sources of random interaction. Maybe you have virtual, non atomic worlds in mind.)

— Such would not be the oscillating universe you envision.

>> A perpetual motion machine as it were. Again, I’m not saying that is the case but I do consider it a possibility.>>

— Not any world causally-temporally connected to or similar to this one, then.

KF

More later, DV — God Willing in Latin.

.

Check your blood pressure JVL. Nothing in your post even made sense.

Who thinks the DNA/RNA/aaRS -mediated replication system was too complex to have arisen spontaneously? Virtually the entire OoL community. That is why they (and you) promote a simple RNA replicator. Can you cite a substantive research program devoted to such a paradigm, where the origin of life

beginswith a DNA/RNA/aaRS-mediated replicator?Of course not.

You already know this, and you knew it when you typed out that sentence.

Upright BiPed: Check your blood pressure JVL. Nothing in your post even made sense.I told you I might have got things wrong.

Who thinks the DNA/RNA/aaRS -mediated replication system was too complex to have arisen spontaneously? Virtually the entire OoL community.I don’t think anyone thinks it arose as it now exists, there would have been precursor stages. Obviously no one knows how it happened but there is no talk of it being guided or created.

That is why they (and you) promote a simple RNA replicator. Can you cite a substantive research program devoted to such a paradigm, where the origin of life begins with a DNA/RNA/aaRS-mediated replicator?No, I don’t keep up with that area. But not being aware of something doesn’t mean it doesn’t exist. I know “RNA world” was a fairly popular hypothesis but that some were thinking there may have had to been something simpler first. I’m just waiting to see what the research turns up. I think it’s far too early to decide if its failed or not.

Of course not. You already know this, and you knew it when you typed out that sentence.I’m just giving you my view based on what I’ve read and discerned from people who I judge to know a lot more about such things than I do. I have no reason to suppose they are being deceptive or are promoting a false paradigm. It may be that orgin of life research turns out to be a total bust but it’s too early to judge.

Kairosfocus: Such a succession for all we know is without limit future-wise, it is POTENTIALLY infinite if so, but at any given now there cannot have been transfinitely remote past stages.I still don’t understand why you accept a potentially infinite future but not a past, especially if the universe is oscillating.

The issue is not to get to the transfinite but to arrive here FROM a transfinitely remote actual past.The positive and negative integers extend infinite far in both directions. But that doesn’t stop us from working with values part of the way along that continuum. IF the universe is oscillating we’re just appearing in one of those oscillations. WE don’t have to ‘get here’ from an infinite past. We are where we are, somewhere along the continuum.

Reality R has some distinct thing, A. What is not A is then ~A. A is a simple unit, ~A is a complex one, different from A. Consider together and duality drops out.But, what does this have to do with our topic? I’m sorry but it seems irrelevant.

Where this leads to is the unreasonable effectiveness of Math in the world and in the Sciences especially. See discussion as already linked:That’s a whole ‘nother topic which I am happy to discuss but I think it should be taken up separately.

A necessary being will be part of the framework for any world that may exist or exists. We just saw it for numbers and it extends through N, Z, Q, R and C too, defining a structure for quantity in any possible world. That gives Math conclusions their powerful generality. By the logic of what being a world is, such must obtain. Math (more particularly a core tied to the Euler identity 0 = 1 + e^i*pi) is as valid in Flatland or Narnia as in our world.— that’s an astonishing result and it gives great power to Physics, Chem, Computing etcI don’t see how the Euler identity makes much of a difference to most working physicists, chemists or programmers. Yes, it can be used when modelling systems as can a lot of other mathematical identities.

Look, when I look at some mathematical structures like Pascal’s triangle or the Fibonacci sequence and they keep turning up in the real world it does seem somewhat miraculous. But a whole lot of really useful mathematics is just messy and complicated. I don’t see any of this implying a necessary being. It’s not a real miracle, it just seems astonishing to us.

God, presumably can add and move around energy. That is beyond the scope of thermodynamics as a going concern world discipline.— Logic of being realities don’t work by energy. The creator is author of energy as Creator.

— Logic of being is not using energy. God as creator put energy where it was in the first place, regulates how it moves about as sustainer and author of laws of nature. If he acts in unusual ways, there is no reason why that should be in principle undetectable.So, God creates the universe and the laws that govern it but is not subject to those laws? So, we were led to believe the universe was lawful because of a law breaking God? I’m confused.

H’mm, there is considerable discussion of a multiverse and what I have suggested is in the first instance open to that. Most obviously in setting the singularity at 0 on a causal temporal succession sequence— There MAY be an at least quasi-physical cosmos beyond our observable gamut. I am open to such.I think there is no evidence yet. Just a lot of guessing. Which is fine, come up with ideas, look for evidence, test things, etc. I’m just highly sceptical at this point.

that is just what would NOT be the case. An oscillating cosmos would be part of a connected, cumulatively acting causal temporal system and energy and energy effects from cycle n would affect circumstances in cycle n+1 etc.Why? I don’t see any clear reason why that would be the case. If the end of each oscillation was another singularity why doesn’t that just wipe the slate clean?

That is, you consider a different world order in which, say, statistics of atoms, molecules and particles . . . the foundations of thermodynamics . . . would be drastically different. (I don’t know how you would prevent sensitive dependence on initial conditions and other sources of random interaction. Maybe you have virtual, non atomic worlds in mind.)I suppose wiping the slate clean each time means you might get widely different results each oscillations but, if the slate is wiped clean, we would have no way of knowing that.

.

pointless, erratic, dissembling

JVL, the difference between a potentially infinite onward succession and an allegedly completed infinite actual past is huge. The process of onward steps may continue onward indefinitely and without terminus, it will at any given attained point be finite but without restriction on further steps. Such is eminently feasible. The — contradictory — claim to have actually traversed the transfinite in finite stage steps thus to have arrived here is demanding a logically impossible supertask, as I showed above and elsewhere. Remember the RW constraint, every stage of the actual past has to have once been the present, causally giving rise to its immediate successor, then again and again to now. That’s why infinite actual past implies transfinite traverse from some negative transfinite stage K’. Notice in that light how one traverses the hyperreal span H –> h and back again by use of 1/x as a catapult function, also how the negatives are created en bloc by imposing additive inverses. Such steps deliberately avoid trying to construct a completed transfinite in cumulative incremental, finite stage steps. Similarly, the set builder approach and notation, e.g. N = {0, 1, 2, 3 . . . } only ever says, here is a stepwise, potentially transfinite process that indicates a set. Notice, for further instance the contrast of transfinite induction vs the common induction proof we are taught in High School; the latter faces the challenge of limit ordinals such as w which do not have a clear incremental predecessor, e.g. there is no specific finite k in N such that k+1 = w. Part of the challenging conceptual leap to grapple with the transfinite. KF

PS: As I have pointed out above, your suggested possible world of an oscillating cosmos not limited by the statistical mechanics that founds the laws of thermodynamics, would be radically different from ours, i.e. without a microstructure and effects traceable to sensitive dependence on initial conditions or other sources of randomness. The only such worlds I can think of would be effectively mathematical exercises and linked simulations. We use them all the time, actually, e.g. a perfect pendulum or rotating system, a perfect sinusoid producing oscillator etc. They are only approximately connected to our observed world. I intend to address more later, RW calls with a delayed LA sitting and load of bills. However, I note, that wiping the slate between cycles of your proposed world actually requires supernatural intervention, i.e. you are proposing a fresh start world every cycle; usual oscillatory models are about dynamical, causal-temporal unity and cumulative effects from one cycle to the next. Hence the challenge of accumulating entropy. Observed flatness means, too, there is not a credible density to lead to a contraction. Instead in recent years, the talk is of accelerating expansion.

JVL,

Let me clip from 30 above, highlighting:

Let me extend, showing beyond n:

. . . . K’ –> K’+1, K’+2 . . . K’+p –> K’+p+1, K’+p+2 . . . . -2, -1, 0, 1, 2, 3 . . . n, n+1, n+2 . . . . w . . . . K –> K+1, K+2 . . . .

(Where K’ + K = 0 and [K+1]’ would lie left of K’ through the logic of additive inverses.)

Notice, numerically, you can define a preceding specifically transfinite range beyond K’ and I am actually indicating that K is far more transfinite than w, first transfinite ordinal. See the surreals construction to see how that can be done. Sorry, this is ever more hairy.

Causally-temporally however, the succession beyond K’ would never reach to some finite span from 0. In fact, it never reaches a close successor to w’s additive inverse. And as this is a representation in the abstract, the root cause is that a preceding transfinite traverse to K’ is not feasible either. We are only warranted to contemplate a finite temporal-causal past, on the logic of structure and quantity of what such a stepwise traverse would require.

There was a beginning to the ordinary temporal causal succession we experience. Even, if we cannot specify just where it is, it is there by the logic of structure and quantity involved.

Onward, we may contemplate a going concern succession without definite terminus, but that is a POTENTIAL not a completed transfinite traverse. Categorical distinction.

And yes, in this lurks a range of meanings for “infinite” and related terms that need to be sorted out.

KF

JVL, as a further point to help with vision, ponder the joint impact of the 1/x catapult from transfinites to the near neighbourhood of 0 in [0,1) and the meaning of continuum. First, [0,1] captures 1/x for the extended number line from 1 on without limit. There are as many infinitesimals as transfinites, but all squashed into the v. close zone around 0 smaller than 1/n for any natural counting number. Second, that on the view of continuum that for ANY r in R — specifically, including irrationals — and ANY close but distinct neighbour s, we may define an intermediate valid number, we may then use addition to slide [0,1) to sit on r and squeeze in between r and r+1. Instantly, the continuum is everywhere utterly infinitely dense, and transfinite continuum cardinality is a class of quantity. Every r in R has a cloud of hyperreals in its extremely close neighbourhood. We can plausibly extend this to R*. Further to this, we now see a way to interpret r + dr, which opens up worlds in analysis and more practically, Calculus. Likewise, the old Geometers were swimming in truly deep waters when they defined points and lines then went on to figures. So too, we can see that in any finite continuous span, there is continuum cardinality, it is a microcosm of the whole transfinite span R* imaged through various operations. The continuum is shockingly strange, more astonishing than we can conceive. I can see some of why Cantor was such a mystic, and perhaps how the weight of thinking so deeply on these topics could have plausibly helped to overwhelm him. KF

Kairosfocus: The — contradictory — claim to have actually traversed the transfinite in finite stage steps thus to have arrived here is demanding a logically impossible supertask, as I showed above and elsewhere.If the universe is oscillating then I don’t see how we can say how old it actually is; only the cycle we’re part of. So we don’t know when the oscillations began so . . .

Anyway, we’ve had our say and I’m happy to drop it now.

Notice in that light how one traverses the hyperreal span H –> h and back again by use of 1/x as a catapult function,I’ve never heard of a catapult function. Are you just referring to its vertical asymptote for some reason?

Similarly, the set builder approach and notation, e.g. N = {0, 1, 2, 3 . . . } only ever says, here is a stepwise, potentially transfinite process that indicates a set.Yes . . .

Notice, for further instance the contrast of transfinite induction vs the common induction proof we are taught in High School; the latter faces the challenge of limit ordinals such as w which do not have a clear incremental predecessor, e.g. there is no specific finite k in N such that k+1 = w. Part of the challenging conceptual leap to grapple with the transfiniteYes . . .

As I have pointed out above, your suggested possible world of an oscillating cosmos not limited by the statistical mechanics that founds the laws of thermodynamics, would be radically different from ours, i.e. without a microstructure and effects traceable to sensitive dependence on initial conditions or other sources of randomness.Like I said: I would assume that the beginning of each oscillations would ‘wipe the slate clean’ so it would be a new start every time, i.e. no lingering effects from the previous oscillation.

RW calls with a delayed LA sitting and load of bills.I don’t know what RW or LA are but I get you’re busy.

I note, that wiping the slate between cycles of your proposed world actually requires supernatural intervention, i.e. you are proposing a fresh start world every cycle; usual oscillatory models are about dynamical, causal-temporal unity and cumulative effects from one cycle to the next. Hence the challenge of accumulating entropy.I don’t see why. Because I don’t see why supernatural intervention has ever been necessary!!

Observed flatness means, too, there is not a credible density to lead to a contraction. Instead in recent years, the talk is of accelerating expansion.Yes, that is true. But they do keep changing their minds!

Notice, numerically, you can define a preceding specifically transfinite range beyond K’ and I am actually indicating that K is far more transfinite than w, first transfinite ordinal. See the surreals construction to see how that can be done. Sorry, this is ever more hairy.I have actually studied mathematics academically so you don’t need to spell everything out on my account. If you want to do so for other readers that’s fine but I won’t respond to every point.

as a further point to help with vision, ponder the joint impact of the 1/x catapult from transfinites to the near neighbourhood of 0 in [0,1) and the meaning of continuum.LIke I said, I won’t respond to all posts since I’m well aware of the mathematics.

JVL, RW again calls, but things have caught my eye. I think you know the Godel result that no comprehensive axiomatisation is consistent, for things comparably complex to “arithmetic.” That is a hard stop for axiomatisation as a guarantee, hoped for, of a coherent system. But, lurking in HS level work is a surprising result that ties the five most powerful numbers in Math together, locked to infinite precision thus utter coherence. The Euler identity. That gives tremendous confidence in the coherence of linked areas. Which, includes a lot of areas closely connected to physics, chemistry and engineering. KF

JVL, I was just off the phone on legislative bills and more, still under pressure. I note, that the logic I pointed out is a hard limit to any purported transfinite in the past sequence of finite stages. All such are dead, period: violation of structure-quantity constraints on being. There is no traversal of the transfinite from actual past stages, that can reach now. If we come here through a succession of before/after ordered stages [notice, this is different from thermodynamic continuity and constraint], there is somewhere a past limit, in effect a beginning. A reality root. The real issue is, what is that root. KF

KF,

Maybe you should submit some more of this, errm, Mathematix™ to johnnyb as a follow-up article. 🙂

DS, I am on the way out now. The constraint on logic of structure and quantity may be unpalatable but it is clear enough. Once we have an ACTUAL past that was once the present, and stages are of finite duration in stepwise succession, traversal of the transfinite is a decisive challenge, failed by all transfinite in the past models of the world we inhabit. KF

Kairosfocus: I think you know the Godel result that no comprehensive axiomatisation is consistent, for things comparably complex to “arithmetic.” That is a hard stop for axiomatisation as a guarantee, hoped for, of a coherent system.I don’t think he said the systems are inconsistent. In fact his proofs only apply to consistent systems.

So, basically, you cannot build a consistent axiomatic mathematical system from the ground up. Sometimes you have to take a step sideways, outside the system.

But, lurking in HS level work is a surprising result that ties the five most powerful numbers in Math together, locked to infinite precision thus utter coherence.I know, it’s a beautiful equation . . . I’m not sure those are the five most powerful numbers in mathematics . . . I’m not sure what ‘powerful’ numbers are . . . He did miss root 2 though which arguably destroyed a whole paradigm of thinking.

I note, that the logic I pointed out is a hard limit to any purported transfinite in the past sequence of finite stages. All such are dead, period: violation of structure-quantity constraints on being. There is no traversal of the transfinite from actual past stages, that can reach now. If we come here through a succession of before/after ordered stages [notice, this is different from thermodynamic continuity and constraint], there is somewhere a past limit, in effect a beginning. A reality root. The real issue is, what is that rootI think we’ve, together, beat that horse past the point of death don’t you?

There isn’t any evidence for any RNA world. There isn’t any evidence that nature can produce RNA’s that could even get a RNA world started.

As for an infinite past, how do we get to the now from an infinite past? Do tell.

ET: As for an infinite past, how do we get to the now from an infinite past? Do tell.How far back do you think time goes? A few thousand years? 14 billion years? What was around before that?

JVL, back just now. I am pointing to the second incompleteness result not the first. Let me cite from SEP, as a handy context:

Hope that helps.

KF

PS: 0, 1, i, e and pi are absolutely pivotal. 0 revolutionised Mathematics. 1 is the start point of number systems. i brings in the complex world, pi is at heart of geometry and things that oscillate or rotate, e brings in too much to enumerate, esp. when joined to i. Draw in sum, multiplication and exponentiation. Infinitely precisely locked together in one expression.

JVL- Don’t try to change the subject. Just answer the question I asked.

As for an infinite past, how do we get to the now from an infinite past? Do tell.JVL, since I am here, I note that length of past timeline is irrelevant; though you will note 0 to now would be 13 – 14 BY on the last values I looked for. The irregular cluster HR patterns with branching from main sequence to Giants bands set that, as would incidence of white dwarfs. The beyond I have addressed is on suggested multiverse models. My basic point is, once we have finite duration stages in succession, the ACTUAL past cannot be transfinite on logic of structure and quantity . . . note the application of mathematics to the criterion that the real past must have been actual. Time, strictly is a within a going concern phenomenon. We have that temporal sequence cannot extend to past infinity and a cosmos does not come from non being. This puts necessary being as the last man standing. Where as from utter non being nothing can come, as a world is, something that is causally independent and adequate to source a cosmos always was. KF

PS: I note, the balance on merits of an issue obviously unwelcome to you is distinct from our opinions. Sorry, once we are dealing with unobservable past of origins and the like we are looking at philosophy strictly, not Science. Doing phil while dressed in a lab coat and using mathematical apparatus does not cut off what it opens up or that the full range of world root consideratins become applicable under comparative difficulties. Besides onlookers now and future will draw their own evaluation and need to do so on a subject where information is hard to find.

KF,

I can’t speak for JVL, but it’s not that your conclusions are “unwelcome”. Rather, it’s that you are knocking over a strawman but refuse to acknowledge it. It’s obviously a disingenuous debating tactic.

DS, it is not a strawman. The suggestion you formerly made kept putting the transfinite traverse as already accomplished at any particular past time; repeatedly begging the key question which is that a transfinite past — whether implicitly or explicitly — will mean the duration from certain particular actual past stages to now would be itself transfinite. That transfinite span is the issue and it cannot be traversed in stages of finite steps. And as someone who has put LIFE repeatedly on the line not merely career, on matters of pivotal truth, I take strong exception to your accusation of “a disingenuous debating tactic.” You are out of order, sir and should withdraw such. KF

PS: Just to be clear as to the objectionable tactic on your part,

For cause, this word as used by you is unacceptable. Kindly withdraw it.

KF,

I’m sorry, but we were not born yesterday.

An argument essentially identical to yours already appears in:

Whitrow, G. J. “On the Impossibility of an Infinite Past.” The British Journal for the Philosophy of Science, vol. 29, no. 1, 1978, pp. 39–45.

The philosopher John Bell points out the fatal flaw in:

Bell, John. “The Infinite Past Regained: A Reply to Whitrow.” The British Journal for the Philosophy of Science, vol. 30, no. 2, 1979, pp. 161–165.

KF,

PS: I’m a little surprised at your reaction. You call our intellectual honesty into question

routinely. You even have tags for your posts involving “Darwinist Rhetorical Tactics”, “Agit-Prop”, etc.DS, turnabout, given immediate and uncalled for context. I will note, that the cases where I have spoken of Agit Prop and Rhetorical tactics, have been cases where there is specific warrant for the view. Unfortunately this is an era of manipulation and when major supporters of a view characterise those who question on terms like: “ignorant, stupid, insane or wicked” there is little choice on the point. Of course, the penumbra of attack sites speaks for itself, down to on the ground stalking. I am rather surprised to see you use such language as you did above. For cause, I object. KF

KF,

In this very thread you have claimed that we made objections to certain points simply because we found the alternatives “unpalatable” or “obviously unwelcome” (reading our minds in the process, no doubt).

Not that such behavior is a capital offense—and this is UD, not a tea party.

Edit: It seems you often a form of the word “turnabout” when someone challenges you on a point, as if it’s somehow invalid to do so. Again, not of great moment, and we’re all used to it, but perhaps suboptimal.

DS,

I took some time to do a search on your papers, I can find the reply but not the original. The reply responds to something I have not argued (assuming this is an accurate summary) so I find it tangential. I am not arguing about Tristam Shandy writing down his daily activities and being slower than his activities but both accumulating to the transfinite.

Nor do I merely assume or assert that one cannot span a transfinite succession in finite stage, cumulative sequences. I SHOWED why.

A big part of our exchanges in the past pivoted on confinement to the reals (with Z interpolated), and I eventually went to Dr Wood on that subject, cf https://uncommondescent.com/mathematics/logic-first-principles-17-pondering-the-hyperreals-r-with-prof-carol-thomas-including-infinitesimals/ . I have concluded, for reason, that R* is a more reasonable model and context for number issues that originally came up with number line, infinitesimal and transfinite discussions that first came up in second to fourth form for me. First in Algebra then in Calculus [when I first carried home 4th form Calculus homework, my Dad said, that at my then age he didn’t know it existed when he was a boy].

I was delighted to find out that Abraham Robinson tamed infinitesimals and thus we can see Calculus foundations as extending Algebra on R*. It was you who pointed me to the Surreals, and I thank you.

The hyperreals allow us to connect naturals, reals, infinitesimals and transfinites in one common number space. The j* operator rotation, vector view extends this to my 6th form experience: Complex numbers. And I recall being in an Astronomy club meeting in QC on St Michaels Row in Bridgetown and having a student ahead of us go through the derivation of the Euler Identity. Over the years since, Y, it has been a wonderful acquaintance.

Now, coming back, it is clear that the ACTUAL, cumulative to present past has to be that, actual. Duration of the past is time [however accounted for] lapsed since definite past stages that were actual. Duration since p to now, n is n – p, on some metric. You know why I speak of stages and sequence counts across such. Call this a common sense stipulation, however you may disagree or agree.

In that context, for cause of question begging, I reject any approach that for any past p accessible to now in a finite number of finite stage steps, infers that the transfinite is beyond p and that we cannot reasonably discuss a past point K’ as I discussed above. I care not that one may say that every number we can count to c, will be at finite remove and so as we may then go c+1 etc we conclude the set {0,1,2 . . . c . . .} is transfinite as a whole defines a transfinite set. What I point out is that the ellipsis is material and that we can better address the matter in R* mileposted with Z* which sets up, again clipping 30 above:

Notice, I have found a way to model an explicitly transfinite span and to show how a stepwise incremental process will not span it, which should be familiar too from the exercises of moving around a given transfinite hyperreal as Dr Wood does in an onward linked video. You can use 1/x to catapult across the span between transfinites and infinitesimals but that is different. In short, the infinity lurks in the ellipses.

In that context, you will observe how carefully I distinguish numbers in finite step counting range from 0 from those lurking beyond transfinite ellipses. In that context putting a preceding further transfinite ellipsis beyond K’ does not affect that duration from K’ to now would span a transfinite range. The above shows that it cannot be done in finite stage cumulative steps.

If one dismisses K’ and points to some c’ on the negative side with a transfinite beyond it that is always given as already traversed, the duration from c’ to now is finite and the transfinite spanning has been question beggingly assumed. It is thus always implicit that a transfinite duration has to be spanned so let’s put it up explicitly.

R* lets us do so and brings out the trouble.

KF

DS, it seems you wish to go off into the weeds on a quarrel, which is a classic context from which master manipulators derived the turnabout projection strategy and cross-complaint strategy; which are proven ways to cloud, confuse, poison and polarise an issue. I will note, when I have pointed to turnabout projection tactics, it has been for cause AS YOU FULL WELL KNOW. Right now you are coming across as how dare you call specific, known, observable tactics all too commonly seen in these dark days of a waning civilisation by their names. Sorry, it does not work that way, especially when you are obviously distracting from going over the line gratuitously. Those we simply do not need, given that from 30 I put up a clear enough reason why I hold for cause as I have again outlined. We may have a disagreement, but in no way can it be reasonably held that it is because of “disingenuous debate tactics” on my part. And, you need to explain to us all, if you object to the conclusion, how we can have a transfinte past duration to now in a causal temporal world without actual, specifically transfinitely removed past points, whether implied or explicitly acknowledged. Duration, p to n is n – p on some metric. The ellipsis is not a past event but an indicator of such. Duration is event to event. KF

KF,

I don’t mean to be especially quarrelsome, rather I’m just making some observations. My jimmies are not the least bit rustled. I’ve had my say on the subject.

PS: I see your post #131 above, and will read it a bit later once I get some work done.

KF,

There is one thing that might be useful to address:

This alleged “question begging” strikes me as very odd. If you’re sincerely proposing this, I have to conclude it’s an extreme case of motivated reasoning.

The infinite past proponents hypothesize that the past can be modeled using the real numbers. Here are a couple of obviously true statements.

Let x be any real number. Then the set (−∞, x) has the following properties:

1) It is unbounded. Given any positive integer M, there is a number p in (−∞, x) such that |x − p| > M.

2) If p is in (−∞, x), then |x − p| < ∞.

Certainly I can say that without begging the question, correct?

ET: As for an infinite past, how do we get to the now from an infinite past? Do tell.You can’t, everyone knows that; I said as much already! But not being able to ‘get to’ infinity doesn’t mean there isn’t an infinite past.

Anyway, I think we’ve beat this topic to death already.

Kairosfocus:I’m perfectly happy with there being a finite past; that’s pretty much how I think about things for everyday purposes: the observable universe began between 13.5 and 14 billion years ago.

Discussing the possibility of there being an infinite past outside of that is just philosophy at this point and I thought it was interesting to pursue for a bit. We’ve both had our say and are starting to repeat ourselves so it seems to me it’s a good time to let it rest. We’ll just stick to our observable universe from now on. We haven’t agreed on the existence of a necessary being though! I don’t think we will but that’s okay.

DS,

I wont’ bother on the continued projection. here, “motivated” reasoning. The pivotal issue is simple enough: what does DURATION since the past mean? Esp. when claimed, infinite.

The real numbers are irrelevant. As is well known, first, Zeno paradox sequences converge as infinite successions in finite time as causally linked increments of space and time form a cumulative, convergent succession. This is tied to foundations of Calculus.

Far more relevant is that our past comes in successive, cumulative finite stages, we commonly use years and seconds. Uniformity and duration are irrelevant, the key issue is successive cumulative effect of the actual past to now. Any ACTUAL past stage was once the present and has been succeeded in sequence to now by further stages.

These can be counted and such a count is a metric. Where, clearly from p n, duration d = n – p, on some reasonable metric; where, it is also well known that the continuum is mileposted with the whole numbers. THERE IS NO DURATION SINCE THE PAST THAT IS NOT A DURATION SINCE SPECIFIC PAST EVENTS.

In that connexion, we can readily see that if duration since p is finite,past since p is finite. It is my contention, here, that appeals to the infinitude of Z or R have in them a pointing arrow or two or ellipses for a reason: they do not have defined borders within the set, once any given z in Z or r in R are given, they can be exceeded, so we do not have a definable limit. However w is order type of the natural counting numbers and the continuum number attaches to any extended span in R^n, including R^1. The pointing arrows on axes or ellipses in set builder notation count.

We can, however capture Z and R in context and that makes a difference, through the hyperreals, R* mileposted by Z*. We then see that there are transfinite spans extending these numbers into the specifically infinite ranges. Mathematically, these transfinite spans cannot be traversed in finite stage successive steps, which is why step builder notation points to the completion of a potential infinity by using ellipses or the equivalent. Where also, the 1/x catapult say prof Carol Wood used allows us to see that from 0 to transfinites, we have a unified space and structure of quantities. Where, infinitesimals and transfinite hyperreals have as much claim to “reality” as Complex numbers, negative numbers, zero, real but irrational numbers such as pi or e etc. We may freely model:

. . . . K’ –> K’+1 . . . . -2, -1, 0, 1, 2 . . . n . . . . w . . . . K-1. K, K+1 . . . ., where 4-dot ellipses denote specifically transfinite ellipses and the continuum is mileposted by numbers in Z* as shown.

Now, let us consider your:

Do you see what comes in in interval indications with (- inf … or . . . +inf)? The ordinary curved brace indicates an OPEN ended interval with no definable terminal element. Here, as there is endless onward continuation L-ward or R-ward respectively. In short the continuum line goes on implicitly to transfinite hypereal spans which cannot be represented within R mileposted by Z explicitly, but we know that any specific given value we can write down or specifically represent can be exceeded, most easily seen with z in Z.

This is actually the issue I found in 2nd form Algebra and 4th form Calculus. It is why on accessing Surreals and hyperreals, I could see that my Math teachers were struggling with the inadequacies of R mileposted by Z as models of the full span of relevant quantity, which became acute with integrals spanning to infinity and with the use of infinitesimals that had been tamed less than a decade before. There was that handwaving struggle with the clumsy epsilon-delta limits approach that while it carefully distinguishes potential from actualised infinity, hides more than it reveals.

Blow the lot up and use the hyperreals and surreals to set context, say I!

I yet further contend that playing within the trainer-wheels of R and Z glides by the lurking hyperreals by using unacknowledged potential, pointing infinities and thus implicit hyperreals.

Combine that with the requirement of actuality of the past and definition that duration since is not between an implicit potential infinity and now, but must only be between definite events/stages that were once the actual present and now. From this, the claim of an infinite past — stripped of its potential, pointing infinities and made explicit — requires actual past events K’ that are specifically transfinitely remote. Which is where the breakdown happens as trying to span a transfinite in steps of finite size is a patent supertask. Hence, again from 30:

There is motivation in my reasoning: to get the matter clear. I find it appalling to hear talk of how a collection of only finite values {0,1,2 . . .} is collectively infinite as beyond any z in N we may go onward z+1 etc. All that says to me is that any number we may successively count up to will be finite but exceed-able with a three dot ellipsis here pointing to the potential but not exhaustively counted infinity. Put it in the broader context and pop, things are clearer.

The difference between normal and transfinite induction is of similar order, especially noting the significance of limit ordinals with no identifiable immediate predecessor, e.g. there is no definable, particular z in N so z+1 = w.

In such a context, it seems to me rather to beg the question to suggest for every L-ward finite z in a succession of stages in a causal-temporal order that reaches now, one asserts or implies that the transfinite succession has already happened. Mathematically we can define the onward L-ward or imply it, but causal temporal succession has to have an effective means and duration has to be anchored to real world actual stages on either end.

As we know all too well from the little dash on a tomb stone between two dates now commonly called sunrise and sunset in my neck of the woods.

I think there is need for some rethinking.

KF

JVL, while too busy to further go into further points in detail just now [forced prioritisation], the remote past of origins is specifically not observable. It is inferred from traces and the logic of temporal-causal succession. Where as I noted long since, the beyond the singularity is commonly raised and must be addressed. Yes phil dressed up in mathematical apparatus and while wearing lab coats is still phil not empirically controlled science. Hence grumblings about strings and vibrating colliding branes, quantum fluctuation sub cosmi and the like. We know that the logic of structure and quantity is a clear guide on what is possible or not and one thing that is not is to have a succession of stages that is a completed actual infinity. That is what needs to be faced, even before thermodynamic considerations are on the table. Back to bills for debate in RW. KF

Kairosfocus: RWRW . . . RW . . . OH! Real World! Whew! That took me far too long!

Are you part of the Legislative Assembly? According to Wikipedia there are only nine members of that body. Is Mr Taylor-Farrell part of the debate as Premier? It seems like he’d still be part of the LA correct?

JVL, advisor-analyst. Mr T-F is current premier. BTW, 9 elected reps, others are ex officio, in a unicameral. HM is designated head. Right now banking definitions unpleasantly remind me of working around pathologies of math. KF

Kairosfocus: Right now banking definitions unpleasantly remind me of working around pathologies of math.Nicely put!!

How is this issue for you

>>Subject to subsections (4) and (5), a licensed

financial institution or licensed financial

holding company shall not, except with prior

written approval of the [Eastern Caribbean] Central Bank, declare,

credit or pay any dividend or make any other

transfer from profits if the licensed financial

institution or licensed financial holding

company realises a net loss for that financial

year. >>

As Divs, stock splits, buy backs etc in loss making years send signals to investment markets, and as profit/loss is up for debate and for revision even post auditor’s statement, plus taxes are often in negotiation for years on end, that means, unless Fed Jr approves IN WRITING, no gravy for investors. I am going to suggest, divs that materially affect required reserves. Banks and shareholders have a legitimate interest in using prudent reserves to damp down instability of share prices and bond values. Also, stock buybacks are implicit dividends and Banks have legitimate interests in stabilising share prices by holding and buying/selling their own shares, contradictory as that may seem. Let’s not get into mutually owning cartels like Canada’s big six.

Then there is this one:

>>in the definition of “affiliate” by deleting

paragraph (e) and substituting the following—

“(e) a company which—

(i) has the same beneficial owner; or

(ii) shares common management,

and has any interlinked business with

F,”>>

It is wider context why I let that one pass.

Then try:

>>in the definition of “banking business”, in

paragraph (a)(ii), by deleting the word

“frequent”; >>

PLUS, in princ amd to Jan 2019:

>>“banking business” means—

(a) the business of receiving funds through—

(i) the acceptance of monetary deposits which are repayable

on demand or after notice or any similar operation;

(ii) the

frequentsale or placement of bonds, certificates,notes or other securities,

and the use of such funds either in whole or in part for extensions

of credit or investment for the account and at the risk of the

person doing such business;

(b) any other activity recognised by the Central Bank as banking

practice and which a licensed financial institution may

additionally be authorised to do>>

[–> this required fiddling with alleged pdf protective schemes, thankfully I did not need to snapshot, OCR and work from text. Lawyers are annoying that way. What logic operator implicitly connects a i and ii? I will suggest an AND operator so the deposit business in some form must be there for the investment side to be part of a bank]

So, is a lawyer or a CFO or realtor who occasionally does such now a banker carrying on a banking business? If so, a whole new order of de facto licencing of fit and proper persons, beneficial ownership, disclosures, reporting and oversight by Fed Jr now obtains. And you betcha that some officious regulator would one day make that case.

Then there’s the push: we gotta get rid of those dirty bills carrying germs.

Do you want a time stamped data shadow of every transaction you do, where, when, in the hands of the ilk of 7th floor FBI bldg and their media slander trumpeters? (KGB, Gestapo, Stasi, hold my three beers . . .)

And more of such order.

Rev 13:

Now, back to mere branes and hyperreals with transfinite durations.

KF

Kairosfocus: >>Subject to subsections (4) and (5), a licensedfinancial institution or licensed financial

holding company shall not, except with prior

written approval of the [Eastern Caribbean] Central Bank, declare,

credit or pay any dividend or make any other

transfer from profits if the licensed financial

institution or licensed financial holding

company realises a net loss for that financial

year. >>

In the words of the Monty Pythons: RUN AWAY! RUN AWAY!!

Anyway: good for you for trying to improve the status of your fellow citizens and guests by slogging away at such bureaucracy. I don’t think I’d have the stomach for it but it certainly needs to be done!

Fight the good fight!

JVL, run away is exactly what I cannot do. This is my “dayjob” stuff. I doubt the regulators spot the issues I am highlighting. And Rev 13 is directly relevant. The ECCB Governor went on radio with a parson — cherry-picked I think — to allay the 7th floor FBI concerns of pushing for cashless. When I wanted to pay a phone bill by cheque in one bank here, I was told not possible, I went to the other . . . which just bought the first [another LONG story] . . . and oh just deposit to A/c xyz. Did. Then now, the local credit union is stepping up near bank operations that leverage that it has different reserve requirements [20% in absolute terms of that for banks proper] though it faces board requirements written in by a previous administration over protest of “a relevant client” on coops based on research including cases such as Kenya. Board members are by implication responsible for any wrong done by any officer, in law, as opposed to there being a negligence or worse standard. Lesson 0, clever glib cutting rhetoric can make sense and caution sound silly, esp when backed by a partisan, morally numb parliamentary majority. Then, to go back and fix unless things visibly go over the cliff is next to impossible given onward firehose flood of new things. I have not been alarmist in pointing to lemmings heading over a cliff, that is a daily observation. KF

KF,

Heh. Sometimes I wonder if this whole interaction is just a crazy hallucination.

Anyway, this is where my “strawman” accusation comes in. You immediately dismiss the claims the infinite-past proponents (IPPs)

actuallyput forth, and start talking about whatyouinterpret “infinite past” to mean. Predictably, it’s very different from what the IPPs mean.At that point, you’re talking about something else entirely.

DS, I gave a particular, specific reason for that, starting with addressing Zeno then showing why we need to set reals in wider context in a unified space of quantities. That included drawing out what open interval notation with infinite sides points to. Potential vs actual infinities are central, please address the substance. The ellipses and simple round brace glyphs, pointing arrow tips on number lines etc are highly significant. You can see what else is specifically on my mind just now. Maybe, that will help to show why I can be so particular and even insistent. KF

PS: Kindly, observe:

Then as directly follows:

Note, I clip and respond.

DS, there is likely an incommensurateness gap of paradigms here, where I am saying limitless continuation points to definitively transfinite numbers and to the challenge of spanning what the ellipses are indicating or round braces on an inf, in finite stage steps. Notice, R and R* are mileposted with integers, broad sense. KF

KF,

As far as I can tell, none of the IPPs who work earnestly on this stuff propose the existence of pairs of time coordinates separated by an infinite interval. Apparently that’s just too far-out even for them.

I don’t see any point in discussing a proposal that no one thinks is plausible in the first place.

DS, so, kindly define a metric or model of duration of actual time lapsed that does not require actual past points in the span to now. Do so, without violence to what time lapsed is generally understood to mean. Do so, without implying what you do not wish to allow. My model is, the past has stages that were once present but were incrementally succeeded to now. So for a past stage p and now n, duration p to n = n – p on some reasonable scale, e.g. years etc. From this, I argue that a transfinite past requires actual past points beyond any finite past point. Further, ellipses, arrows, open ended inf intervals give a pointer to the transfinite, they don’t actually express it. A set can be without finite limit, a duration requires at least representive end points. I find we keep shunting off the infinite into, already traversed by any given p. KF

JVL:

You don’t have to get to infinity. You have to get to the now from an infinite past.

ET: You don’t have to get to infinity. You have to get to the now from an infinite past.No, you can’t be at an infinite past. You can only be at some point along the way.

Infinity is not a place.

In the physical realm, infinity is an absurdity.

You aren’t anywhere (not a place or time or anything) in an infinity.

Andrew

ET & JVL, the actual past has to be what was actually once the now, succeeded incrementally and causally-temporally to the current now 2020 AD. If that now past is infinite there were once nows from which the current now would have been the transfinitely remote future; and I don’t care if there was an onward further infinity, it is duration K’ to today or p to today that spans the transfinite that have to be explained. KF

AS, you raise a point, the issue is an actually traversed infinity, Some are trying to get that without the supertask of spanning the infinite in finite cumulative steps. They are trying to use R and Z to do it, I am pointing out the fuzziness — transfinite fuzziness — of the borders of those sets and how R* and Z* mileposting it lets us see more clearly. When that is there, duration since p to n = n – p brings out no ends of trouble. KF

KF,

Eh? I’m pretty sure that the IPPs (some cosmologists, I gather) do not propose such a thing. Of course there are actual past points in the, errm, past.

All I can say is, it hasn’t worked. Apparently people have decided that in reality, this is not true.

I don’t know what the state of the art is in infinite-past cosmology. All I can do is random googling. Here’s the abstract from a document on arxiv.org which might help in understanding what these people are saying:

Kairosfocus: If that now past is infinite there were once nows from which the current now would have been the transfinitely remote future; and I don’t care if there was an onward further infinity,That ‘now past’ could not be at infinity. Infinity is not a place or time or value. You can only ‘be’ at times a finite distance away from the present.

JVL, if the actual past is infinitely large, given causal temporal succession there were once nows from which the current now would have been remote beyond any number of stages n in N. Otherwise the span from those past points p is finite and bounded by 2021 to come. KF

JVL:

I didn’t say that you could. I understand that an infinite past in impossible. You were the one talking about an infinite past.

Not with an infinite past.

DS, we can use whatever notations we will, the issue is to make coherent sense of a claimed infinite causal-temporal past without implying a span that cannot be completed in finite stage steps. See my just now to JVL. Gotta go out now. KF

ET: I didn’t say that you could. I understand that an infinite past in impossible. You were the one talking about an infinite past.There can be an infinite past but you can’t be at infinity. Just like the number line: there are infinitely many integers and you can ‘be’ at any one of them. But you can’t be at infinity.

Not with an infinite past.That’s how it works, just like the number line. Think of today as being zero on the number line. Tomorrow is 1, the next day is 2 and so on. Yesterday was -1, the day before was -2 and so on. You can be at 27 or 356 or -389994567473737 but you cannot ‘be’ at infinity.

You don’t ‘get’ from infinity to finite because you can’t ‘be’ at infinity. It’s not a place or value or time.

Kairosfocus: if the actual past is infinitely large, given causal temporal succession there were once nows from which the current now would have been remote beyond any number of stages n in N.Nope, any ‘now’ you specify is a particular point or time along the time line. It’s not infinity. It will always be some finite number of steps away. You cannot be infinitely far away, you can only be a finite number of steps away, so you never have to worry about getting from the infinite to the finite.

Otherwise the span from those past points p is finite and bounded by 2021 to come.No, because any limit or span can be exceeded very simply. There is no limit or span for an infinitely large set. That is the point after all.

If you specify a now or time or value that’s finite. There is no finite value that traipses into the infinite.

JVL,

You are having conceptual problems. If you can be at a point, then you have switched to a finite context.

Andrew

Asauber: You are having conceptual problems. If you can be at a point, then you have switched to a finite context.Think of the number line, extending infinitely far in both directions. You never get to the end, to infinity, there is no end. Infinity is not a place or point. You can’t ‘be’ at infinity.

If you specify a particular point then you are specifying how many finite steps away from zero, the centre, you are. You cannot be infinitely far away at any given step even though there are infinitely many steps.

I don’t know if there is an infinite past but you can’t dispute it using this kind of mathematical argument.

JVL

“Think of the number line, extending infinitely far in both directions.”

This doesn’t help your problem. Infinity isn’t comprehensible. It does nothing for anything you want to demonstrate. You just put it there for arguments sake. When you consider a point, you have changed the context to something comprehensible, which is finite.

Andrew

Asauber: This doesn’t help your problem.I think you missed the beginning of the conversation. All I’m doing is disputing

Kairosfocus‘s mathematical reasoning why there cannot be an infinite past.Infinity isn’t comprehensible. It does nothing for anything you want to demonstrate.I think it is comprehensible. There are different sizes of infinity and some ways of handling them. Get a Set Theory textbook if you’re interested.

You just put it there for arguments sake. When you consider a point, you have changed the context to something comprehensible, which is finite.Of course, which is why I find

Kairosfocus‘s reasoning incorrect. He says there is a past ‘now’ that crosses the line into the transfinite which is not correct.JVL,

“I think it is comprehensible. ”

An idea of it is imaginable, I guess, like a square circle, but nothing that could ever resolve to any usable information. Again, details are finite. Like a point is.

Andrew

Asauber: An idea of it is imaginable, I guess, like a square circle, but nothing that could ever resolve to any usable information. Again, details are finite. Like a point is.It’s used in mathematics all the time. First place you’d most likely see it is in Calculus but it’s really hammered out in Set Theory which gets pretty weird eventually.

“It’s used in mathematics all the time. ”

JVL,

I know. Abstractly.

Andrew

As 164: I have put on the table the hyperreals which do make sense of transfinite values and infinitesimal ones. Only, they do not help the case of those who want a transfinite temporal causal past traversed in finite steps. KF

JVL 161:

Did you really mean to say:

That is what I am saying about the actual past as opposed to the abstracta of setting up R* with Z* as mileposts.

KF

JVL, 163:

Which THE number line? The hyperreals allow you to get to transfinitely removed values readily. Take the 1/x catapult function, feed in a suitable h smaller than 1/z for any natural counting number, yes, an infinitesimal. 1/h –> H, a number in excess of any z in N. Where too, h is familiar for foundational calculus, Newtonian fluxions style:

f'(x) = lim as h –> 0 of [f(x+h) – f(x)]/h

What you cannot do, however, is to start at 0 and take steps, 1, 2, 3 etc and reach to H.

That supertask cannot escape a finite span from 0 though it can point to a potential infinity, where having counted to any particular k, we can now go k+1, k+2 etc and that could be put 1-1 with 0,1,2 etc. We see that the naturals and reals do not have a crisp border but go on and on. However there is no definable specific u so u +1 = w, the first transfinite ordinal.

Yes, as there are transfinitely many transfinite integers in Z* you cannot be at infinity, but you very well can be at a transfinite H which is at infinite remove from 0. Those are matters of structure and quantity bound up in the framework for any possible world. Necessary beings that shape and constrain what is possible in any world. And see, that concept is important, applying to the universality of core mathematics. It is not an arbitrary matter of opinion.

The arrows on a number line drawing point to the transfinite realms.

In this context, time as considered in chaining finite stages [e.g. years] is causal-temporal and successive. 2019 –> 2020 –> 2021. We can label the stages and will see that the logic of being property of successive finite stage steps then imposes the challenge of the supertask of trying to span the transfinite in steps.

KF

Kairosfocus: As 164: I have put on the table the hyperreals which do make sense of transfinite values and infinitesimal ones. Only, they do not help the case of those who want a transfinite temporal causal past traversed in finite steps.I will say this again: you cannot move from the infinite to the finite in finite steps. No one is saying that. Any moment in the past is a finite number of steps away. You are never ‘at’ the infinite.

That is what I am saying about the actual past as opposed to the abstracta of setting up R* with Z* as mileposts.I’ve said what I’ve said. If you consider the timeline like the number line and we are at zero on the number line then time stretches out to infinity before and behind us. At any given moment in time we are a finite number of steps away from today. You NEVER move into the transfinite or the hyper-reals or any of that.

Which THE number line?The real number line.

The hyperreals allow you to get to transfinitely removed values readily.Yes but that’s not the point. Today is time zero on the time line. Tomorrow is 1, the day after is 2, etc. Yesterday is -1, the day before is -2, etc. The timeline stretches out infinitely far before us and after us. But every step along the way is a finite number of steps away from today. You do not need to talk about reaching H or whatever. You don’t get to infinity. You can’t. I’t’s not a point on the line.

Take the 1/x catapult function, feed in a suitable h smaller than 1/z for any natural counting number, yes, an infinitesimal. 1/h –> H, a number in excess of any z in N. Where too, h is familiar for foundational calculus, Newtonian fluxions style:f'(x) = lim as h –> 0 of [f(x+h) – f(x)]/h

What you cannot do, however, is to start at 0 and take steps, 1, 2, 3 etc and reach to H.And no one is saying you can!!

That supertask cannot escape a finite span from 0 though it can point to a potential infinity, where having counted to any particular k, we can now go k+1, k+2 etc and that could be put 1-1 with 0,1,2 etc. We see that the naturals and reals do not have a crisp border but go on and on. However there is no definable specific u so u +1 = w, the first transfinite ordinal.That’s right, you can’t get to the infinite in a finite number of steps. There are no ‘potential’ infinities. They are or they aren’t.

Yes, as there are transfinitely many transfinite integers in Z* you cannot be at infinity, but you very well can be at a transfinite H which is at infinite remove from 0.No, not in our analogy. You cannot be at some infinitely distant point. You can only be at some finite point. You’re trying too hard to defend your position and your muddling up the maths.

Those are matters of structure and quantity bound up in the framework for any possible world. Necessary beings that shape and constrain what is possible in any world. And see, that concept is important, applying to the universality of core mathematics. It is not an arbitrary matter of opinion.I don’t know if there is such a thing as a necessary being but I know your mathematical argument is not correct.

The arrows on a number line drawing point to the transfinite realms.They point to continuing on ad infinitum. Forever. That is not a place or a being.

In this context, time as considered in chaining finite stages [e.g. years] is causal-temporal and successive. 2019 –> 2020 –> 2021. We can label the stages and will see that the logic of being property of successive finite stage steps then imposes the challenge of the supertask of trying to span the transfinite in steps.You can’t span the transfinite in steps! You can’t be at infinity.

I’m not sure we’re getting anywhere here. I am not arguing about your philosophy or theology, just about your use of mathematics which I think is incorrect.

JVL, to propose an actual, infinite past is precisely to imply if not explicitly declare, spanning the transfinite in finite stage steps. As to why, that has been drawn out above and in onward linked. KF

KF,

No. That is all.

PS: To clarify, it does not imply ever being at a point at infinite remove from the present.

I’m afraid I have to award this round to JVL.

DS, explain the actual duration. KF.

KF,

Each interval between any two points in time is finite.

The set of durations of these intervals is unbounded.

It’s just like the statements I posted above about intervals in the set of real numbers.

DS, so far as I can see, all you are doing is leaving the transfinite traverse implicit and possibly lurking under a contradiction or two. I can note that if all traverses between stages of time are finite, that says something regarding the global set, if that were meant literally. And, it says something about the evolutionary materialistic worldview that it is forced to commit itself to a past-infinite material world (including some pretty bizarre claims to make such seem half plausible), which should give pretty serious pause to would-be materialists and sympathisers. KF

PS: It’s been quite a day.

Kairosfocus: And, it says something about the evolutionary materialistic worldview that it is forced to commit itself to a past-infinite material world (including some pretty bizarre claims to make such seem half plausible), which should give pretty serious pause to would-be materialists and sympathisers.I am NOT committing myself or anyone to an infinite past, I am merely pointing out that you could have one an NOT be transversing the transfinite. I am disagreeing with your use of mathematics.

I am happy to consider that time began with the big bang and leave it there.

JVL (& attn DS),

the hesitancy to go there speaks. There is something uncomfortable with the appeal to the structure of R mileposted by Z.

That’s where the problem I have been pointing to lies.

Before going to that, I note on why of old the Steady State vision of a cosmos that had an undetectably low steady origin of matter etc was preferred to where the empirical evidence forced things in the ’60’s to 70’s. It has no beginning, in effect the root of reality would be the cosmos, so reducing the first law of thermodynamics to an approximation was a good fit to the dominant preference. It just did not fit the data.

The hard beginning at the singularity implies a contingent, caused world, given the challenges of a cosmos from utter non-being. It is unsurprising to see models that move to multiverses, quantum foams, brane collisions etc. All of which move beyond empirically controlled science and all of which are philosophical. Which on comparative difficulties cannot be provincial; factual adequacy is global, coherence is global, explanatory elegance and power are global. For instance, the logic of being is relevant, the moral government of our rationality is relevant [thus, the IS-OUGHT gap], the roots of functionally specific complex organisation and associated information is relevant, observed fine tuning is relevant, and more. Dressing up in the lab coat and posing on evolutionary materialistic scientism backed by selective hyperskepticism will not do.

On many of these grounds, the evolutionary materialistic world picture falls apart as utterly incoherent, factually inadequate and forced into ever wider ad hoc patchwork.

Coming back to the narrower question, Mathematics is about the logic of structure and quantity, i.e. it has deep roots in logic of being. In ontology, in short. Key abstract entities and structures are necessary beings, part of the framework for any possible world. Which is what gives it its power, answering Wigner’s wonder.

Now, let us look at Z as mileposting R. Clipping 146 above (and beyond):

The span in the middle sets R and Z in wider context. Where, I remind, the 1/x catapult function between infinitesimals and transfinites secures a unified number space. There is no gold standard special threshold R and Z pass that R* and Z* don’t, reducing them to second class and hyperskeptically suspect status. Recall, [0,1] has in it the catapulted image of every number from 1 on, with the infinitesimals near 0 being a cloud — monad is a term used — closer to 0 than any 1/z, z in N. The continuum is exceedingly strange. Those infinitesimals are a legitimate foundation of the Calculus, once suitably tamed. If infinitesimals like h are valid and part of the framework, so are transfinite hyperreals such as H = 1/h, period. Addition allows shifting the cloud of hyperreal infinitesimals, revealing that any r in R is surrounded by a similar cloud to 0. In effect, the number line is a vector space, once negative numbers give another direction and extension. Extension to C is instant, through the j* rotation operator. And of course, that cloud around an arbitrary real, is part of how calculus gains its power.

The hyperreals pervade the reals and are everywhere incredibly dense. And by the similarity principle for R*, the same pattern extends to the transfinite realm. The continuum is exceedingly strange.

This little exposition brings out part of what I think is going on. Paradigm shift. We are used to thinking in terms of N, Z, Q, R, maybe C. Now, I am arguing that R* pervades R everywhere with incredible density, giving fresh force to the idea of the continuum. This, we are decidedly not used to, the inextricable intertwining of R* and R, connected to the power of Calculus. The expression r + dr takes on new meaning once we see that dr is an image from the infinitesimal cloud around 0 vector shifted to r’s close neighbourhood.

Paradigm shift, r* is everywhere dense with infinitesimal-altered numbers in R and beyond of course. And not even mind bleach is going to take us back to the “safe” old days before the R* pandemic. R* is the new normal. Vouched for, by the power of Calculus. BTW, reducing an infinitesimally altered number in R* to a standard value in R looks uncommonly like taking a limit as dr –> 0; i.e, going to the core of the cloud around a particular real. A further intuitive bridge to familiar territory, here, the epsilon, delta close neighbourhood, limits picture as now influenced by the infinitesimally altered number picture.

We need those pictures as plausibility structures. For sure, I needed them back in 4th form and arguably in 2nd form as we looked at asymptotes on graphs and on those pointing arrows on number lines. In College, the idea of an open terminus of an interval with no definable specific closest neighbour was also obviously connected. It also shows the echo of that transfinite ellipsis that I have chosen to highlight by using a 4-dot symbol not the usual 3. In a continuum, borders are fuzzy not crisp unless you insert a Dedekind cleavage, a schnitt. (I gather the word tends to a stronger suggested force in German.)

Let us hold that fuzzy cloud in the small picture, vector shift it back to 0 and use the 1/x catapult to see its dual in the large. The border between the transfinites and the finites is . . . fuzzy, with no nearest definable finite neighbour to the transfinites and the converse. Oddly, there is a parallel in history. In the old days, borders of great empires were not crisp, surveyed lines on a map [formerly fought over by competing Kings and reflecting the emerging technology of precise survey], but graded zones of influence. Here, a review on a book on borders of the Roman Empire:

Paradigm shift, fuzzy structures are real, and the reals are fuzzy, not just in the cloud around a safely finite real, but in the large, gradually and undefinably fading out as we come to the transfinites. This reflects the phenomena of the surreals — isomorphic to certain developments of the hyperreals — whereby we can go on stepwise to develop a complex structure of numbers around a given transfinite. Similarly, for hyperreals specifically. So, the idea of counting up in steps from K’ to K’+1, etc is not unreasonable and the result that it is a supertask to try to span from there to finites in steps reflects the fuzzy zone. (Recall, K’ is in the additive inverse region, negatives.)

In that context, a further point of paradigm shift.

Namely, that when we speak of z in Z as mileposting R then showing how any particular z can be exceeded R-ward z+1, or its additive inverse z’ can be exceeded l-ward, we have crispified and so have yet to address the onward fuzzy zone to the transfinites. That is what the pointing arrows and ellipses are telling us, also the round braces marking open ends of intervals. Crispification does not mean the fuzzy zone is not there, it is pervasively present as we look to the zone where finites go on and on with the recognised transfinites on the other side.

It is, in my view, the want of reckoning with that fuzziness that leads to the idea that crispification by count from zero or by simple algebraic symbolisation — which brings in count up from zero so soon as one does substitution instance — allows us to effectively disregard the implicit onward structure of the fuzzy zone. The ellipses etc are part of the structure of Z and R. They point to the fuzzy zone, and with the suffusing or pervading of R by infinitesimally altered reals, R and R* simply cannot be separated anyway. We have a unified, inherently transfinite space and structure of quantities, one embedded in the framework for any possible world.

It is from this different paradigm that I am seeing.

In this context, when we insist on crispifying durations n – p, we are inherently on the finite side of the fuzzy zone, we have made the infinite character implicit. Consequently, I suggest a modified way of thinking in light of the unified vector space: for any defined past instant or stage p that can be connected to n, now, here 2020, in a sequence of cumulative finite stage steps, n – p := d, duration since p, will be finite. However, the very act of crispification locks us in on this side of the fuzzy zone and the interconnexion of stepwise, finite stage access locks in that we are only warranted to talk of finitely remote p’s. The idea that we can capture the fuzzy onward span L-ward by saying “any p” becomes a conceptual leap too far.

In short, it seems that the very structure of R is locking us into talking of finitely remote points once we crispify, but the onward fuzzy zone is material.

That is part of what is exercising me, the frustration of the receding fuzzy zone as we count outwards or as we crispify by symbol that represents a finite count.

This is a key part of why I think the way we have seen discussed above fails to capture the challenge of an alleged transfinite past. I think crispification implicitly, perversely, subtly locks us on this side of the fuzzy zone. The fuzzy zone is material to the beyond crisp limits character that conveys the countable transfinite character of Z mileposting R.

In this context, I further believe and argue for cause, that once we make the transfinite character of duration as claimed, we then are on the other side of the fuzzy zone and cannot cross it in finite stage steps to reach a finitely countable range of 0.

The structure is frustrating the attempt, not a misconception.

Hence, why I think we are talking on two sides of a paradigm shift.

I believe that, implicitly, intuitively, inchoately, by 4th to 6th form, I was on this side of the paradigm. The bringing in of Surreals and Hyperreals simply gave me a better apparatus for expressing where I had been for decades. Maybe the best symbol for that was definite integrals with infinite limits. Those, were crying out for a number space that allowed us to make sense of that. R* and its cousin, the Surreals, have given me that space.

From that perspective of an intellectually fulfilled hyperreals thinker on the structure and logic of quantity, I can make sense of the puzzles from those days, once I took on board the 1/x catapult and of course the vivid image of the hyperbola. We can christen it, the hyperbolic catapult.

In addition, the concept of duration, d = n – p, allows me to see why I have been so uncomfortable with attempts to speak to a claimed infinite past with all symbolised past points p comfortably finitely remote. The fuzzy zone, perversely, is being inadvertently locked out by the implicit finitisation by algebraic symbol.

Which then has consequences for how one interprets the matter at stake.

Further, that inadvertent lockout is why I maintain that the reference to K’ is legitimate, giving d = n – K’ (defined on a stepwise count metric) an inherently transfinite character and pointing to why once the implicit lockout is recognised, we are not warranted to suggest a transfinite actual past that accumulates stepwise to now, in finite stages.

All of this then becomes part of the worldviews exchanges. I do not expect agreement across paradigms, I do think there is a right to suggest that a “gold standard” lockout, however, is not warranted.

KF

Kairosfocus:It’s *R by the way, not R*. I can’t imagine you seeing anything of hyperreals except at University, maybe even graduate school, unless you just read it yourself. It’s certainly not standard to make these kinds of arguments in undergraduate Analysis courses.

I think we’ve exhausted this topic to be honest. Like I said: I am NOT saying there is an infinite past, I am not proposing an infinite past, I am saying that I didn’t find your mathematical argument against it sound. That’s it.

I don’t think continuing the discussion any further will substantially move things forward. Let’s call it a day shall we?

JVL, both are used. I used it the other way in my paper. I do not actually expect you to agree . . . the polarisation is too predictable for that. What I have done is to show that there is a serious problem, that evolutionary materialism has serious implications in terms of ultimate origins, that an assumed infinite origin is not a default, and that there are serious issues about the transfinite, that are tied to ever so many routine things. For cause, what I find unsound is the idea that in a causal-temporal cosmos, with cumulative final stages to now there is an idea that we have an infinite past with durations to now from every past stage being finite. This directly implies global finitude. Instead, I find that at each occasion we specify a particular p that is finitely removed in steps from now, the onward fuzzy zone is where the transfiniteness resides. Further, duration is between specific times and if the finitude of the past remote point in question — it matters not if there are yet further onward stages — is relaxed, the gap cannot be traversed in finite stage steps. That leads me to conclude, there is no warrant for claiming a transfinite actual past accumulating stepwise to now. Especially, if one also claims that all past points are only finitely remote. I suspect, the implicitness of the fuzzy zone commonly represented by an ellipsis, is lurking unacknowledged, with force much as I outlined. KF

KF,

So now there is a cabal of “””elites””” locking the hyperreal numbers out of physics?

Here are my hypotheses: (1) Physicists/cosmologists/etc. simply find it unnecessary to use the very weird and complicated hyperreal number system, when the relatively tame set of real numbers suffices. Or, (2) that reformulating physical theories in terms of the hyperreal numbers would be a heckuva lot of work, where there’s no indication anything is to be gained.

Here’s a paragraph from a post on the physics stackexchange concerning a related issue. We all know random posts on stackexchange are always true of course:

Kairosfocus: What I have done is to show that there is a serious problem, that evolutionary materialism has serious implications in terms of ultimate origins, that an assumed infinite origin is not a default, and that there are serious issues about the transfinite, that are tied to ever so many routine thingsI’ve said, several times, that I am NOT trying to push forward the ideas of a possible infinite past; it is not even my belief, It’s not an issue with evolutionary issues. Why do you keep trying to make it so?

For cause, what I find unsound is the idea that in a causal-temporal cosmos, with cumulative final stages to now there is an idea that we have an infinite past with durations to now from every past stage being finite. This directly implies global finitude. Instead, I find that at each occasion we specify a particular p that is finitely removed in steps from now, the onward fuzzy zone is where the transfiniteness resides. Further, duration is between specific times and if the finitude of the past remote point in question — it matters not if there are yet further onward stages — is relaxed, the gap cannot be traversed in finite stage steps.Exactly right, the gap cannot be traversed in finite steps. And any given point in the past is a finite number of steps away from the present.

That leads me to conclude, there is no warrant for claiming a transfinite actual past accumulating stepwise to now.And I didn’t say there was!!

Especially, if one also claims that all past points are only finitely remote. I suspect, the implicitness of the fuzzy zone commonly represented by an ellipsis, is lurking unacknowledged, with force much as I outlinedI’m suggesting we just leave it there and stop beating the horse past death. Really, it’s time to quit. I am not going to agree with you no matter how often your repeat the same thing. You are not going to agree with me. So let’s stop.

Here are my hypotheses: (1) Physicists/cosmologists/etc. simply find it unnecessary to use the very weird and complicated hyperreal number system, when the relatively tame set of real numbers suffices. Or, (2) that reformulating physical theories in terms of the hyperreal numbers would be a heckuva lot of work, where there’s no indication anything is to be gained.Or, maybe, you don’t grasp the mathematics as well as you think you do.

Just let it go. It doesn’t even matter that much. I’ve agreed to consider that our measurable time starts about 13.5 – 14 billion years ago. That is the materialist viewpoint.

Why do you keep going on and on about it? I do not understand.

DS,

there is a clearly ideological domination in our civilisation, evolutionary materialistic scientism. For instance, the Nobel Prize winner Monod:

That carries over into Physics as a matter of course, but that is so just about everywhere.

More on point, I have suggested on experience and observation, that the hyperreals are more what we deal with, what with infinitesimals. Which BTW tend to be used rather freely in Physics as a holdover from C18 – 19 early. I used to actually mutter about mathematical trickery as a somewhat hybrid student, partly in that context.

>>Here are my hypotheses: (1) Physicists/cosmologists/etc. simply find it unnecessary to use the very weird and complicated hyperreal number system, when the relatively tame set of real numbers suffices>>

I already mentioned the rather free use of infinitesimals, even in improper differentials such as in d’q/T in the second law of thermodynamics. I used a dash for the actual bar. Hyperreals are all over the place in Physics and related fields where infinitesimals — oops, differentials — also get used as algebraic fractions.

>> reformulating physical theories in terms of the hyperreal numbers would be a heckuva lot of work, where there’s no indication anything is to be gained. >>

The hyperreals are already there as infinitesimals and infinitesimally altered quantities, copiously there. What would be needed is a bit of

justificationand advice on not getting into C18 type troubles.Your stack exchange quote may relate to renormalisation etc. My concerns start with the free use of infinitesimals, as noted.

So, this tangent fails.

KF

JVL, attempted piling on fails. KF

Kairosfocus: JVL, attempted piling on fails.I have no idea what you are talking about.

KF

Translation: ‘Two or more people disagree with me.’

KF,

Ok, I must have misunderstood. The hyperreal numbers are indeed not being locked out.

DS, I never claimed they were locked out of physics; it should be clear from above that from 2nd to 4th form High School, issues tied to R* were involved in Math-related issues I encountered, and all the more as Calculus came in. Since you brought it up, they were deeply embedded in College Physics, but there was a problem of loose C18 – 19 style use without insight from the [then fairly new, admittedly] C20 developments. I am also simply interested in opening up the number space as early as reasonable, much along JB’s lines. Where there is an issue is as came out on your intervention above. KF

PS: Some reading https://arxiv.org/pdf/1205.0174.pdf

KF,

You can certainly do calculus with infinitesimals if you like. If it works for the physicists, that’s great. For undergraduate math students, I don’t think it should be a high priority. Maybe for the more advanced students, who can understand the construction.

DS, In the context of physics, my concern is to

moderatewith the lessons learned by Abraham Robinson and others, also to put into the quiver an understanding of the full range of available quantities and structures. Then, it wouldn’t hurt, to focus a bit on logic of being roots on why Logic of structure and quantity — Math — is so useful (per Wigner), perhaps even including a bit of possible worlds thinking, setting up a more conscious approach to relevant modelling. All of which said, it is interesting that one result of the above is a drawing out, more and more, of the sort of worldview level commitments that are incrementally required to take up and keep evolutionary materialistic scientism and/or fellow traveller views. I am very aware that P => Q can always be responded to, ~Q so, not P, but then at worldviews level, there are not that many root-level options and the cumulative commitments are sounding to me a lot like Cycles and epicycles etc c 1500 – 1650. KFPS JB’s textbook is high school level, in effect; I guess in the US system, Advanced Placement. In recent years I ran across a UK GCSE Math textbook. Basic Calculus is now in what would have been ordinary not additional math in my day. Many years back, I learned that in Russia, every High School kid had to do 4 – 5 years of Calculus level Math and Physics, deemed necessities for the common defence.

EG (JVL): Actually, when there is a me too pattern without substantial cogent contribution, especially in support of an argument that may itself lack cogency, piling on becomes a question. KF

F/N: In a sense we are full circle. The Godel argument pivots on logic of being and on the need for adequate, necessary being world root. It is somewhat abstracted from issues such as, rooting of responsible, rational freedom and associated inescapable, built in moral government. It is known to be of valid formal structure (at least in the adjusted version), and along with other similar arguments on this and other subjects, it is premises that what have to be challenged, to reject conclusions. Thus, we see the real issues: P => Q, ~Q so ~P carries with it issues of what alternative premises and what are their difficulties. Also, what does ~Q itself entail. So, we are back to issues of worldviews choice, comparative difficulties analysis and what conclusions are we willing and able to live with? Why? KF

PS: Such issues become cumulatively challenging for evolutionary materialistic scientism and fellow travellers.

For example, it is manifestly inescapable that our rationality is governed by duties to truth, to right reason, to prudence, to sound conscience, to neighbour (esp. fairness and justice) etc. We are under built-in moral government and also need to credibly account for rationality and the ability to warrant sound knowledge.

In that context, immediately, any worldview that undermines rational, responsible, morally governed freedom is in the end futile. Where, reduction of rationality to computing on a GIGO-driven substrate confuses ground and consequent, freely taken inference governed by first duties of reason, with blindly mechanical and/or stochastic, cause-effect processes driven by the organisation of a substrate is patently self-referentially incoherent. That already decisively undermines evolutionary materialistic scientism and its fellow travellers.

Taken another way, such views try to ground mind in what is grossly inadequate, computation.

Similarly, the inescapable first duties confront us with the challenge of freedom: a free creature is necessarily, free. Thus the is and the ought diverge and choices (sometimes thorny ones) must be made. However the IS-OUGHT gap now confronts us, which, post Godel, can only be resolved and bridged at the reality root.

Such then turns into the issue of roots of a causal-temporal order. Utter non-being is a non-starter. Circular causation where a not yet future reaches back retrocausally is much the same [and yes, there are those who suggest and imply such]. This means there are two serious options, the root is an infinite regress of the material, causal-temporal order, or else there is a finitely remote completely causally adequate reality root that is non-physical as a necessary being.

A subsidiary claim is that necessary being is a dubious or arbitrary construct. I trust the cross-world reality and consequent rational power as well as applicability of numbers suffices to show otherwise. Somewhere, there is a reality root causally independent of external cause.

Much of the thread then turned into a retracing of some aspects of the exchanges over several years. Plainly, an explicitly transfinite span cannot be traversed in finite stage steps, more or less almost by definition of a countable transfinite. The real issue is, are we dealing with an implicitly un-spannable transfinite that cannot be exhausted by similar steps? I suggest that the fact that beyond any n in Z, there are n+1, n+2 etc equally matched to 0,1,2 etc carries that weight. Similarly, that if between any past stage p we can traverse from to now q, the duration d = q – p will be finite, so if all traverses are finite, we have a finite span. If the body of traverses is collectively transfinite, at least one past point K’ will be specifically transfinitely remote.

(The claim that Z in R has in it an endless onward supply of finite values and that defines its infinitude, to my mind only shows that we have the sort of fuzzy zone I described earlier.)

Similarly, to avoid issues of cumulative entropy and finite past [and future] oscillations, rather speculative and patently fine tuned . . . thus credibly contingent . . . observationally unconstrained models of cosmological origins have had to be constructed. For instance, Wiki summarises:

What is clearly happening is that the ~P and ~Q are beginning to bite. There is no credible account for the morally governed mind to come up with such speculations, indeed, rationality and responsibility would reduce to grand delusion. The commitment to a frame in which infinite causal-temporal regress lurks in the suggested implicit infinitude of numbers [where every one we can count to or represent is finite and bounded] opens a further can of worms. The associated neo-oscillating models for cosmology are openly speculative, at best.

This is sounding like the ever growing patchwork that led to the collapse of the Ptolemaic cosmology 400 years ago.

Faith, not founded on essentially indisputable facts of science, and beginning to be a degenerative ideological programme.

Where of course, such speculative philosophy with mathematical apparatus done while wearing lab coats is not empirically constrained science. It therefore faces the full force of comparative difficulties, and it simply will not do to a priori rule out the possibility of purpose as “unscientific,” as we saw with Monod.

Then, the logic of necessary being speaks: a serious candidate necessary being is either impossible in the way a square circle is, or else it is actual. This is the heart of modal ontological reasoning. So, evolutionary materialistic scientism and fellow travellers either need to show God not a serious candidate or else impossible of being. On the latter, the usual argument of 50 years past is undermined, alleged contradiction in the face of evil. On the former, the bill of goods for reality root for a world with morally governed creatures should give pause even if one is dismissive to the sheer raw fact of theism as a major world view position. (Where, if so many are written off as delusional, it undermines the general credibility of the mind.)

So, we have some pretty serious choices to make regarding worldview first plausibles regarding the root of reality.

F/N2: Associated, I note on the quantitative structure of R*, including R in context.

We start with countable numbers mileposting R, where between any n and n+1 we can wedge in a vectorially displaced [0,1), anchored on n+0 = n; recognising the continuum. We get Z by the additive inverse, n’ + n = 0, identity element for +.

Here, there is no

definablelast positive integer n_final such that n_final + 1 –> w, first transfinite ordinal. There is a fuzzy zone as n goes ever higher, where every n we can represent or explicitly count to is bound onward by n+1, n+2 etc. So we can match that to 0,1,2 etc.This is related to how we recognise the transfinite nature and it is why the ellipses in Z := { . . . -2, -1, 0 1, 2 . . . } do a lot of work, too often not sufficiently recognised.

Also, incrementation fails to attain the transfinites, we exploit continuum in [0,1) and the hyperbolic catapult 1/x to recognise an infinitesimal h where h is less than 1/n for any n in N, and in turn 1/h gives a hyperinteger H greater than any n in N. (We here have chosen h strategically.) This wider span then opens up similar extensions and additive inverses, yielding the transfinite spans of the hyperreals. Infinitesimals of various character may readily be constructed algebraically on h etc. We can also see that the infinitesimal cloud around 0 can be exended to any r in R by addition, and can similarly cloud around H etc.

We have a unified number space, extensible to the complex domain etc.

I think this domain is useful in itself, just as a unified structure of quantity. It also obviously extends to the Calculus. Indeed, I have seen the suggestion that reducing an infinitesimally augmented result to standard form per Robinson is closely comparable to the limit on the now more conventional approach. My comment here is it may help in forming core calculus concepts which extend to standardised results.

It is in this context that I pointed out that finite means just that and that the transfinite span cannot be traversed in steps. Even, when it is implicit and fuzzy.

KF

PS: Notice on the general issues of cosmology https://uncommondescent.com/intelligent-design/at-astronomy-mag-is-the-big-bang-in-crisis-rob-sheldon-responds/ I note, the HR plots of star clusters with breakaways from the main sequence heading to the giants bands and relative absence of White Dwarfs point to the same general t-frame as expansion, so that ~ 14 BY is a plausible, observationally anchored age for a clearly expanding cosmos. Beyond that, the incrementally built models over the past 50 years seem to be in trouble.

KF,

Would you be able to give a precise definition of this “fuzzy zone” in the hyperreals?

DS, I already pointed out the structural feature that the zone TO the recognised transfinites is fuzzy (reflected in the significance of the ellipses and the n and onward problem where you cannot exhaust to some n_f so n_f + 1 –> w), similar to the fuzzy zone to the infinitesimals and even the open border side of an interval in a continuum such as [0,1). I made a comparison to the old Roman borders where a Wall was built on defensible ground and forays and influence beyond faded out as one went beyond a zone of feasible control — in some ways it is a historical pity they could not sustainably move beyond the Rhine to a shorter river line, but that is a perennial challenge of continental Europe; the fall of the W Roman Empire cost us centuries of development. Back on subject, I am of course bearing in mind the related but not identical idea of fuzzily bordered, partial membership sets. Given part-ways to a full integration or differentiation operation, even operators it seems can have fuzzy borders. So, the ellipsis is material. What we can explicitly count to or represent is finite but always with an elliptically loaded beyond. Surge forth here and the Barbarians retreat beyond reach, then when you pull back, they return. KF

PS: Where h is a value LT 1/n for any specific n in N, such that in the interval (h,1] there is no definable closest neighbour to h, and yet there are valid numbers everywhere in [0,1], including in [h,1], then we see that by catapult, beyond any specific n in N, there is a valid CONTINUUM with milestones that cannot be counted exhaustively all the way to H = 1/h, and particularly we may identify on w as order type of {0,1,2 . . .} that there is no identifable n in N, n_f so that n_f + 1 –> w. Thus, w is a limit ordinal on N, and this points to transfinite induction etc. So, while not a precising definition, we can characterise the fuzzy zones by this sort of process.

Edit: I see the update now.

There is a diagram on the wikipedia page here at the top right. I can’t directly link to it. Can you describe where these zones lie on it?

DS, a key one is just next to 0, of course, the real home of the infinitesimals. A second one will be between some arbitrarily high but specific n in N and w/2 [etc] as is shown in the use of the ellipsis, another will lie between w/2 and w as we cannot count up in steps from w/2 to w or down in steps from w to w/2 and there are many, many more. Indeed, I am reminded of the point that continuum on [0,1) raises the interesting point that h is in the continuum but is not regarded as a real. So, do we hold that there is a forbidden zone and hidden tiny gap . . . one as dense with points in R* as the whole transfinite realm . . . just away from 0 where the infinitesimals would fit, or do we reckon instead that the hyperreals allow a better model of the continuum, one well fitted to say Calculus? Or do we banish the infinitesimals and their hyperbolic catapult 1/x duals the transfinites, holding that there is no valid number beyond what n in N can milepost? The more I look at it the more central R* seems to be, and the more significant those odd puzzles in 2nd to 4th form were. KF

KF,

The issue I see is that all “zones” in the hyperreals are equally fuzzy, as I understand the meaning of that word.

In the real numbers, any two nonempty open intervals (a, b) and (c, d) can be put into one-to-one correspondence via an order preserving function. The two intervals “look” exactly the same except for the labels on the numbers.

I believe* the same holds in the hyperreal setting. Imagine standing x1 = 0 and at x2 = 1/ε. Then the neighborhoods around these two points look exactly the same, in a way that can be made precise (I think).

*I don’t have a reference for this, but will look for one later.

DS, notice the issue on the continuum vs the gap next to 0? A fuzzy gap? And yes, the continuum has an order, howbeit, that order does present the fuzzy zones whenever finite to transfinite or finite to infinitesimal transitions occur. And, it is too late to get rid of those. KF

DaveS: In the real numbers, any two nonempty open intervals (a, b) and (c, d) can be put into one-to-one correspondence via an order preserving function. The two intervals “look” exactly the same except for the labels on the numbers.I was going to say: Good luck with that!

But then I figured out a way to do if for any open intervals. I think.

Anyway, carry on.

DS, yes, by secondary extension, a cloud of infinitesimally altered numbers clusters around each real, which fits with Calculus. The point is, the hyperreals are pervasive once they are in the door. However, the zone to 0 and that as the naturals-mileposted reals run away to the transfinite are primary. My sense of what the continuum is, is thus affected, and of course any material zone of a continuum gas continuum cardinality, with the reals line giving a definite ordering. KF

KF,

Er, no? Not that I’m aware of. Perhaps your #206 is a clarification or correction?

Anyway, it would seem we can dispense with expressions such as “fuzzy zone”.

DS, the principal fuzzy zones are there, such as -1 —-*0* —- 1 and that as the finite values traverse to transfinite ones, we can represent (* . . . -0-1-2- . . . *). The one at: *0* can then be vector displaced all along the line, showing the interwoven nature of R and R*. As well of course: (r-1–*r*—r+1) for any r in R. However, I had pointed to the continuum as we go v close to 0. With hyperreals not in view, we would not see it but there is an issue once infinitesimals of order h are in view. Where the hyperreals are not about to go quietly into the good, good night. So, it looks like we really do need to ponder the interwoven nature of hyperreals and reals, with every real subject to infinitesimal augmentation by displacing a copy of *0* as, r + *0*, Things have got rather hairy, on reflection and R* seems in some ways more fundamental and comprehensive. KF

😳

I have to admit I can’t make head or tail out of the last part of #200 or #208.

DS,

do you not see that once the hyperbolic catapult 1/x is present, and once the infinitesimal h is admitted, using notation I think goes back to Newton, then surrounding 0 we have a cloud of infinitesimal values, *0* which are explicitly not reals but are somehow on the continuum of numbers in the interval [0,1) on the RHS part and (-1,0] on the L-ward side? If there are numbers legitimately in the continuum but that are NOT reals, is there not, then, a fuzzy gap in the reals as asterisked? As in the v close neighbourhood of 0.

As this is a zone, and as the continuum number line is a vector space, we can shift it R/L-ward by tip to tail and infinitesimally alter any given r in a similar way to 0, just “add” the members of the cloud to r. That means R is pervaded, riddled everywhere with similar gaps, once the primary one is established.

The continuum number line if limited to unaltered reals is then sieve-like, with tiny fuzzy gaps essentially everywhere.

Or else, we accept the smearing out of reals, r –> r + *0* –> *r*, and so the continuum as a whole is hyperreal, if it is to be “truly” continuous. This already shows the notion of a continuous number line is in a sense independent of the set, R [taken as excluding infinitesimals and other numbers mileposted by Z but infinitesimally altered, i.e. r + dr is here not a real, allowing reduction to “standard” form]. Once it is so taken, the truly continuous number line is hyperreal. Not, real.

And if we insist R is truly continuous, then it looks like infinitesimals and infinitesimally altered numbers would have to somehow be brought in. Nonstandard reals, I suppose.

Then, we see 1/h –> H, unified with the real span mileposted by Z. The same tip to tail smearing can be applied to H, so we have *H* too. This means the continuum goes to transfinite realms and we have infinitesimally altered transfinites. H + dH would be a valid part of the transfinite span, just that H is transfinitely remote from 0 and 1/H –> h. Do we dare say, nonstandard transfinites? I dunno, this is following the logic as it comes up. I find myself stunned by the idea of doing a Calculus move in transfinite realms.

Coming back through the catapult, if we insist R is truly continuous then infinitesimals are locked in it seems, in which case given a unified gamut of numbers, transfinites knock at the door. So, are the reals truly the continuum mileposted by Z? Where do h, w, H and co fit in? [It is 150 years too late to lock them out.]

Weird.

The fuzzy zones seem to have teeth.

KF

PS: a related discussion https://boxingpythagoras.com/2017/09/30/infinitesimal-calculus-1-the-numbers-between-numbers/ note too the wheel paradox illustration pointing to a continuum cardinality.

Sorry, folks. No matter how hard you try, you won’t be able to make this thread transfinitely long.

PPS: This at Quora gets wild, bringing in surreals perspectives: https://www.quora.com/Does-constructing-the-hyperreal-number-system-require-rejecting-the-axiom-of-completeness-If-not-why?share=1

BO’H, the real issue is, can it be made continuous? See the Quora link. KF

DS, a nonstandard, infinitesimally altered real would be a number not reachable by a power series of rationals that exclude an infinitesimal tail . . . effectively it cannot be written as a whole + place value decimal continued to w steps. In that sense pi + dr is ontologically different from pi, it is a hyperreal in the pi-cloud *pi*. Where, as the tail dr is unique, reduction to standard form of course applies, i.e. the cloud *r* is inseparably attached to r. Hence, r is a standard real limit. This gives teeth to how dr is the catapulted dual of a transfinite. Truly, truly, truly weird but somehow wonderful, I am beginning to see why there was such an intensity in the infinitesimals debate. The smeared out real, *r* looks a lot like the figures can be made from infinitesimal line segments. Jump to the complex numbers, turn this into a 2-d cloud and voila, plane figures are patterns played with these vector-tip clouds. Multiple dimensions beckon. Do you recall, path integrals,curvy line length results and how fields come in through path integrals in physics . . . part of that uneasy trickery I spoke to. KF

KF,

I think I get it (partially) now. I’ll reflect my understanding of the first paragraph back to confirm:

Referring to the diagram on the hyperreal wikipedia page, the pink “number line” on the lower left shows that in the hyperreals, the point where 0 lies in the real number line has been replaced by what looks like an entire real number line, where the units are marked as multiples of ε.

This means, for example, that (−ε, 0) and (0, ε) are intervals of positive length in *R which contain no real numbers. That is, these are “gaps” in R, if we view it sitting inside *R.

****

I would caution that there are no “gaps” in the real number continuum, however, when we view it in its own standard topology. In fact, in some ways, the set of hyperreals has “holes” in it which do not exist in the reals. For example, there are Cauchy sequences in *R which do not converge.

KF,

Continuing as I have time:

When we think of R as being embedded in *R, it is spread out, with “gaps” in a sense. But R under its usual topology has no gaps.

It wasn’t until we messed with R by replacing each point with an uncountable interval that these gaps appeared.

DS, it gets more and more curious. The problem is, R is no longer alone. And, of course, infinitesimals lurk given Calculus. KF

KF,

If by “lurking”, I assume you mean that we can do calculus using infinitesimals if we choose. But it’s strictly optional.

DS, they are there in the concept space, it isn’t going to go away. KF

KF,

There’s no need for them to “go away”. They just aren’t necessary for doing calculus is all.

Ds, the point is, the genie is out of the bottle, once transfinites are there and the link to infinitesimals. . KF

F/N: I add, the hyperreals give us a way to unify numbers from infinitesimals to transfinites on a common scale with counting numbers, integers and the reals they milepost. That in turn lends teeth to the point that a transfinite span can legitimately be shown by using ellipses on a number line and that such cannot be spanned in finite stage steps. Where, too, we see that two primary fuzzy zones appear as the onward count of the naturals mileposting the reals transitions somehow into transfinites and as 1/z in N fails to get as close to 0 as h = 1/H, H a transfinite hyperreal. The price we pay is that on this extended line of quantities — an ordered set and structure — the presence of infinitesimals implies that Reals cannot strictly bridge to 0, one must go to infinitesimals to complete the continuum there. And as the infinitesimal cloud on 0, *0*, can then be added as a vector cloud to any r in R or any transfinite H, this similarly pervades the expanded number line. Where of course once two directions are present in the number line, we have 1-dimensional vectors: size, direction and continuously variable magnitude. I am thinking, this brings in troubling questions as to just how complete the Reals are; the hyperreals look more fundamental to me, though they are challenging. R now takes its place as finite, non infinitesimal numbers mileposted by N. All of this then feeds back into debated issues. The things that lurked in 2nd to 4th form Math have turned out to be crucial. I suspect, inchoately, we were playing with R* not R, and in College physics, it was more than playing with. Mind you I am now finding revisions on C17 – 19 Math that argue that the early greats were not as loose with infinitesimals as has been alleged. Going to standard part looks a lot like taking a limit with hyperreals, admittedly ticklish with convergence issues. How do you define a sequence of partial sums and ever tightening zones in what are already transfinite members of a sequence? Do enough infinitesimals pile up explosively, given exponentially transfinite transfinites? Etc, just feeling around half blind here. Zeno’s revenge? KF

KF,

My answer: The real numbers are even more complete than the hyperreal numbers in a sense. They satisfy every definition of completeness on this page, while the hyperreal numbers do not.

For example, in the real numbers, every nonempty set which is bounded above has a least upper bound or supremum (that’s one form of completeness). For example, the least upper bound of (−1, 1) is 1. That’s the smallest real number which is greater than or equal to every number in (−1, 1).

The hyperreal number system does not satisfy this property. I think you’re using the notation *0* for the set of all hyperreals which lie at most an infinitesimal distance from 0. This set does not have a least upper bound.

1, 1 − ε, 1 − 2ε, 1 − 3ε, and so on are all upper bounds for *0* but there is no smallest one.

DS, that’s the fuzziness zone in action, once the ultra small is in the door given we cannot finitely bound 1/z, z in N mileposting R, as there is no definable last z_f so z_f + 1 –> w. Where not even w counts as we can conceive of w/2 etc. I begin to appreciate why the older mathematicians viewed both infinities with horror; and, continuum, thou art a real headache. They REALLY didn’t have tools to handle such. KF

PS: I don’t mean forming the set basis for a field etc, but covering the range of valid numbers, even bound ones that are in that sense finite, exceeded by other valid numbers. And yes, the infinitesimal cloud is meant, h and relatives closer to 0 than 1/z for any finite z in N. Or 1/r for the interleaved reals. Mileposting counts.

Kairosfocus:There are no gaps in the reals: pick any two real numbers and you can ALWAYS find another one in between them. The Reals are everywhere dense.

the presence of infinitesimals implies that Reals cannot strictly bridge to 0,You can get as close as you want with the Reals.

Remember: Calculus was put on firm ground with limits which do not involve infinitesimals.

KF,

Well, I guess we can continue jamming more stuff in there until it’s utterly incomprehensible 😛

But already, with the hyperreals, we have:

* this weird nonconstructive ultrafilter step

* different choices of ultrafilter apparently may produce different hyperreal fields (I don’t know if this is completely settled)

* Is there a practical way to represent hyperreal numbers for purposes of computation? With the real numbers, there is a unique non-terminating decimal (or binary, or whatever you prefer) representation that is very intuitive. It’s easy to implement libraries to perform arithmetic on (certain finitely -describable) decimal expressions. I haven’t found such a thing for “the” hyperreals. I’ve seen suggestions that this is possible using quotients of polynomial expressions, but I haven’t found any hyperreal libraries for common programming languages.

* Finally, that diagram at the top right of the hyperreal wikipedia page could be extended “above” and “below” indefinitely, I believe.

If we consider the normal real numbers, the corresponding number line looks like:

−2, −1, 0, 1, 2,

multiplying each number by ε, we end up with:

−2ε, −ε, 0, ε , 2ε , (or *0*)

Multiplying by ε once more, we get an even more infinitesimal neighborhood of 0, perhaps **0**:

− 2ε^2, − ε^2, 0, ε^2, 2ε^2,

and so forth. If we multiply by ω = 1/ε repeatedly, we move up the diagram, where the “mileposts” now are in multiples of ω^n for integers n. So we have an infinite hierarchy of neighborhoods of 0 (and of every other hyperreal), going up and down.

This is great fun for a mathematician, I’m sure. Is there any point for a physicist to prefer *R over R, however?

KF (& cc to JVL)

This wouldn’t have been a problem if we had just stuck with R and hadn’t bolloxed it up. 🙂

JVL, once hyperreal transfinites and their duals under 1/x as hyperbolic catapult are on the table, the situation changes; this has been discussed several times above, hence the discussion of fuzzy borders and neighbourhoods. In particular we have h and H as mutual images under 1/x such that h is closer to 0 than 1/z for any z in the counting numbers {0,1,2 . . .} and H is (hugely) larger than any z so delineated. We then have the cloud I for convenience designated *0* — taking in additive inverses — and by vector addition a similar cloud surrounds any z or even any r in R mileposted by Z. Indeed, [0,1) and (-1,0] can similarly be extended between mileposts. So, once infinitesimals such as Newton’s h or Leibniz’s dx are in play things are not quite the same. That problem will not go away. The hyperreals are there, the surreals beyond are there.

DS, The problem would be that R would effectively become R* once transfinites are on the table and once infinitesimals were rehabilitated by what, the 1940’s. I think the line of numerical quantities is what is truly foundational, and R is not an adequate model for its full span.

KF

KF,

I think that R would remain R, regardless of what developments take place externally. We don’t have to tart it up by augmenting it with whatever anyone places “on the table”. It’s a free country, so you can do whatever you please, but I have yet to see a compelling use case for the hyperreal numbers in the empirical sciences. That’s the challenge, in my view: Find a situation in the sciences where *R is indispensable.

If you are just interested in the mathematics of *R, that’s fine, but of course there’s no end to the crazy mathematics we can dream up.

DS, that zone *0* is not beyond N, and is around the mid point of R. KF

PS: Part of my problem is precisely, the free use of *0* in the physical sciences.

KF,

Physicists certainly use infinitesimals informally. And for them, it’s the right tool for the job. They’re doing physics, not mathematics.

However, I have yet to see a case where a computation (in the physical sciences, say) involving infinitesimals cannot be replaced by a computation involving only “standard” quantities (no infinitesimals, no infinities).

That is, I haven’t seen that *R is indispensable for the empirical sciences. Do you know of any cases where it is?

Edit: This post on stackexchange is interesting.

A quote from a physicist in the thread:

(I actually enjoy reading the results of that “wasted effort” so I don’t agree that it’s completely wasted, but from a physics perspective that makes sense).

DS, actually, they are used; famously the second law of thermodynamics, in the Clausius statement giving dS GTE d’Q/T, which then feeds into the wide apparatus of Thermodynamics then combines 1st and 2nd laws which is a very powerful expression and in many other places. Yes, small increments similar to economics are also used. KF

PS: Look here too https://physics.stackexchange.com/questions/92925/how-to-treat-differentials-and-infinitesimals note:

>>There is an old tradition, going back all the way to Leibniz himself and carried on a lot in physics departments, to think of differentials intuitively as “infinitesimal numbers”. Through the course of history, big minds have criticized Leibniz for this (for instance the otherwise great Bertrand Russell in Chapter XXXI of “A History of Western Philosophy” (1945)) as being informal and unscientific.

But then something profound happened: William Lawvere, one of the most profound thinkers of the foundations of mathematics and of physics, taught the world about topos theory and in there about “synthetic differential geometry”. Among other things, this is a fully rigorous mathematical context in which the old intuition of Leibniz and the intuition of plenty of naive physicists finds a full formal justification. In Synthetic differential geometry those differentials explicitly (“synthetically”) exist as infinitesimal elements of the real line.

A basic exposition of how this works is on the nLab at

differentiation — Exposition of differentiation via infinitesimals

Notice that this is not just a big machine to produce something you already know, as some will inevitably hasten to think. On the contrary, this leads the way to the more sophisticated places of modern physics. Namely the “derived” or “higher geometric” version of synthetic differential geometry includes modern D-geometry which is at the heart for instance of modern topics such as BV-BRST formalism (see e.g. Paugam’s survey) for the quantization of gauge theories, or for instance geometric Langlands correspondence, hence S-duality in string theory.>>

Things are complicated.

PPS: Following up (while one ear is on the LA debate) I find https://ncatlab.org/nlab/show/synthetic+differential+geometry

>>

In synthetic differential geometry one formulates differential geometry axiomatically in toposes – called smooth toposes – of generalized smooth spaces by assuming the explicit existence of.infinitesimal neighbourhoods of pointsThe main point of the axioms is to ensure that a well defined notion of infinitesimal spaces exists in the topos, whose existence concretely and usefully formalizes the wide-spread but often vague intuition about the role of infinitesimals in differential geometry.In particular, in such toposes E there exists an infinitesimal space D that behaves like the infinitesimal interval in such a way that for any space X [element of] E the tangent bundle of X, is, again as an object of the topos, just the internal hom TX := X^D (using the notation of exponential objects in the cartesian closed category E). So a tangent vector in this context is literally an infinitesimal path in X.

This way, in smooth toposes it is possible to give precise well-defined meaning to many of the familiar computations – wide-spread in particular in the physics literature – that compute with supposedly “infinitesimal” quantities.

Remark 1.1. As quoted by Anders Kock in his first book (p. 9), Sophus Lie – one of the founding fathers of differential geometry and, of course Lie theory – once said that he found his main theorems in Lie theory using “synthetic reasoning”, but had to write them up in non-synthetic style (see analytic versus synthetic) just due to lack of a formalized language:

“The reason why I have postponed for so long these investigations, which are basic to my other work in this field, is essentially the following. I found these theories originally by synthetic considerations. But I soon realized that, as expedient ( zweckmässig ) the synthetic method is for discovery, as difficult it is to give a clear exposition on synthetic investigations, which deal with objects that till now have almost exclusively been considered analytically. After long vacillations, I have decided to use a half synthetic, half analytic form. I hope my work will serve to bring justification to the synthetic method besides the analytical one.” (Sophus Lie, Allgemeine Theorie der partiellen Differentialgleichungen erster Ordnung, Math. Ann. 9 (1876).)

Synthetic differential geometry provides this formalized language.>>

In short, this issue is live.

KF,

The Clausius inequality makes sense as a statement involving “pretend” infinitesimals. No one can actually make a measurement of an infinitesimal physical quantity.

bio of Urs Schreiber:

Hmm, this guy looks more mathematician than physicist You know how trustworthy

theyare…🙂

However, point taken. The Schreiber post could be an example.

I have great reservations about this statement he makes:

This subset

hasto be just {0}, because there are literally no other such numbers in R. In another space “like” R in some ways there could be others, sure.DS, history. Classical thermodynamics is a continuum theory, it does not depend on the reality of atoms or molecules. This is part of why it has a significant independence of stat mech, which definitely is about populations of particles.Going beyond, differential geometry —

Wiki: >>Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry [–> emphasis seems to be on in the ultra small. see here: http://alpha.math.uga.edu/~shi.....iffGeo.pdf the opening is oh so ever suspiciously familiar!] The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques endemic to this field. >>

— is apparently its own development. KF

PS: The dx^2 = 0 as defining threshold or order scale of infinitesimals etc stuff is familiar, it is some of what go to standard form addresses. And we are seeing fuzzy zones and line segments in continua etc. Some bells are a ringing.

KF,

Yes, differential geometry is essentially calculus on arbitrary manifolds. You can use infinitesimals or not, depending on preference.

PS: That’s “classical” differential geometry I’m talking about, btw. The modern stuff is incredibly abstract.

DS, bottomline, this stuff is not about to go away and once we have ultra small scale smooth domains the catapult concept, suitably applied, points to the transfinites in a unified space. R and extensions to multidimensional spaces, R^n, does not readily accept such. Of course, the popping up of intuitionistic logic with things fuzzy between is itself not without issues. I note some axioms definitely start with domains where h^2 –> 0 as an axiom in that context. KF

KF,

Whew, that’s a relief 😛

More seriously, I don’t think anyone is trying to make this stuff go away, whatever that would mean.

I do think you have developed a tendency to demand that models involving the real numbers also include infinitesimals and infinite numbers, even when the, *ahem*, designer of that model has stated otherwise. They’re “on the table”, so excluding them from the model is clearly a sign of suspect ideological commitments.

DS, no, I am thinking, once we recognise ultra-small and ultra large as well as the in between as a unified structure of quantities, the number line continuum now stands as prior to sets we have used and our set choice has been relativised. Where, infinitesimals in particular have been haunting Math since classical times, not to mention once Calculus provided a unifying general approach. And, I am not so sure we get to invent fundamental quantities that are arguably going to be present in any possible world . . . part of their power. KF

KF,

That’s an audacious move. 🙂

Haven’t you already argued that the set of real numbers is present in all possible worlds?

DS, as part of core quantitative structure, which includes more. KF