Logic & First Principles, 4: The logic of being, causality and science

We live as beings in a world full of other concrete entities, and to do science we must routinely rely on mathematics and so on numbers and other abstract objects. We observe how — as just one example — a fire demonstrates causality (and see that across time causality has been the subject of hot dispute). We note that across science, there are many “effects.”

Such puts the logic of being and causality on the table for discussion as part 4 of this series [ cf. 1, 2, 3] — and yes, again, the question arises: why are these themes not a routine part of our education?

The logic of being (ontology) speaks to possible vs impossible entities, contingent ones and necessary ones. As was tabulated last time:

We see here, that some proposed things such as a square circle are not possible of being: there is no possible world in which they would exist as core characteristics [squarishness vs. circularity] stand in mutual contradiction. This already shows LOI and LNC in action, the principle of distinct identity truly is of central importance. Other things are possible of being (some of which are actual). Where, we may contrast that some must exist as part of the framework for any world to exist (e.g. numbers) — necessary beings. Other things may exist in certain possible worlds but would not exist in others, hence: contingent.

Contingent, on what? Causes.

But first, just what is a possible world? (Is it another name for the multiverse?)

Let us use a surprisingly helpful introductory discussion at Wikipedia — after all,on the principle of charity we must encourage them towards doing a consistently good job:

For each distinct way the world could have been, there is said to be a distinct possible world; the actual world is the one we in fact live in . . .  There is a close relation between propositions and possible worlds. We note that every proposition [–> statement that is true or else false] is either true or false at any given possible world; then the modal status of a proposition is understood in terms of the worlds in which it is true and worlds in which it is false. The following are among the assertions we may now usefully make:

• True propositions are those that are true in the actual world (for example: “Richard Nixon became president in 1969”).
• False propositions are those that are false in the actual world (for example: “Ronald Reagan became president in 1969”). (Reagan did not run for president until 1976, and thus couldn’t possibly have been elected.)
• Possible propositions are those that are true in at least one possible world (for example: “Hubert Humphrey became president in 1969”). (Humphrey did run for president in 1968, and thus could have been elected.) This includes propositions which are necessarily true, in the sense below.
• Impossible propositions (or necessarily false propositions) are those that are true in no possible world (for example: “Melissa and Toby are taller than each other at the same time”).
• Necessarily true propositions (often simply called necessary propositions) are those that are true in all possible worlds (for example: “2 + 2 = 4”; “all bachelors are unmarried”).[1]
• Contingent propositions are those that are true in some possible worlds and false in others (for example: “Richard Nixon became president in 1969” is contingently false and “Hubert Humphrey became president in 1969” is contingently true).

The idea of possible worlds is most commonly attributed to Gottfried Leibniz, who spoke of possible worlds as ideas in the mind of God  . . . . Scholars have found implicit earlier traces of the idea of possible worlds in the works of René Descartes,[3] a major influence on Leibniz, Al-Ghazali (The Incoherence of the Philosophers), Averroes (The Incoherence of the Incoherence),[4] Fakhr al-Din al-Razi (Matalib al-‘Aliya)[5] and John Duns Scotus.[4] The modern philosophical use of the notion was pioneered by David Lewis and Saul Kripke.

We may summarise, a possible world is a description of the way the — or, a — world might be, inter alia requiring coherence and sufficient completeness for purposes of analysis or action.

For example, in mathematics we routinely construct axiomatic systems that lay out complex abstract model worlds even though, post-Godel we know that no  sufficiently complex scheme can be both utterly complete and coherent. Also, that for such schemes we cannot construct an axiomatic system that is demonstrably coherent. (That is, in the end, our confidence in the coherence of our mathematical systems is supported rather than demonstrated; i.e. inductive reasoning is inescapably involved in the practice of mathematics.)

And yes, the utility of Mathematics and its application through computing systems is never far from the surface in our ongoing considerations. Where yes, that means that to some extent we must accept the sufficient reality of a host of abstract, logic-model worlds that we may apply them in our thought and even practical work. Ponder, here, how natural numbers lead to the panoply of transfinite numbers:

We may thus proceed to understand causes and causal factors, first in a fairly narrow sense:

where a contingent entity A would exist in world W1 but would “just” not exist in a closely neighbouring world W2, the difference in circumstances

C(W1 – W2) = f1

allows us to confidently identify f1 as among the relevant causal factors that enable A to be.

Then, we may explore across several neighbouring worlds W2 to Wn, identifying a broader cluster of factors {f1, f2, . . . fn} such that they are each necessary for and are jointly sufficient for A to be.

As an example, ponder the extended fire triangle, the fire tetrahedron:

We are also seeing here the significance of experimental studies, observational studies, use-cases, Monte Carlo modelling and statistical investigations, where in effect we set up micro-worlds and study properties as we vary circumstances or ponder natural variations.

In this context, we are already clarifying cause.  Wikipedia is again helpful for convenience, broadening our view:

Causality (also referred to as causation,[1] or cause and effect) is what connects one process (the cause) with another process or state (the effect), where the first is partly responsible for the second, and the second is partly dependent on the first. In general, a process has many causes,[2] which are said to be causal factors for it, and all lie in its past (more precise: none lie in its future). An effect can in turn be a cause of, or causal factor for, many other effects, which all lie in its future. Causality is metaphysically prior to notions of time and space.[3][4]

Causality is an abstraction that indicates how the world progresses, so basic a concept that it is more apt as an explanation of other concepts of progression than as something to be explained by others more basic. The concept is like those of agency and efficacy. For this reason, a leap of intuition may be needed to grasp it.[5] Accordingly, causality is implicit in the logic and structure of ordinary language.[6]. . . .

Causes may sometimes be distinguished into two types: necessary and sufficient.[14] A third type of causation, which requires neither necessity nor sufficiency in and of itself, but which contributes to the effect, is called a “contributory cause.”

Necessary causes
If x is a necessary cause of y, then the presence of y necessarily implies the prior occurrence of x. The presence of x, however, does not imply that y will occur.[15]
Sufficient causes
If x is a sufficient cause of y, then the presence of x necessarily implies the subsequent occurrence of y. However, another cause z may alternatively cause y. Thus the presence of y does not imply the prior occurrence of x.[15]
Contributory causes
For some specific effect, in a singular case, a factor that is a contributory cause is one among several co-occurrent causes. It is implicit that all of them are contributory. For the specific effect, in general, there is no implication that a contributory cause is necessary, though it may be so. In general, a factor that is a contributory cause is not sufficient, because it is by definition accompanied by other causes, which would not count as causes if it were sufficient. For the specific effect, a factor that is on some occasions a contributory cause might on some other occasions be sufficient, but on those other occasions it would not be merely contributory.[16]

J. L. Mackie argues that usual talk of “cause” in fact refers to INUS conditions (insufficient but non-redundant parts of a condition which is itself unnecessary but sufficient for the occurrence of the effect).[17] An example is a short circuit as a cause for a house burning down. Consider the collection of events: the short circuit, the proximity of flammable material, and the absence of firefighters. Together these are unnecessary but sufficient to the house’s burning down (since many other collections of events certainly could have led to the house burning down, for example shooting the house with a flamethrower in the presence of oxygen and so forth).

Obviously, a specific contingent circumstance — e.g. the unfortunate burning down of a specific classmate’s house on a particular day in 1976 — had particular distinct causal factors summing up to its specific cause (of interest to the Insurance company) etc. However, once we loosen to a house burning down, we see that we can properly take cause in a looser sense (e.g. of interest to those writing fire safety regulations). This them makes good sense of sufficient but not necessary causal factors as for instance Mackie raised.

Obviously, for an event E, all necessary causal factors (in this looser sense) must be present, as knocking out any one will block it. Likewise, a sufficient cluster must be present which may include broader contributory factors. For example, while a court building here could have caught fire through a short, fire fighters and the police were very interested to observe evidence of accelerants. Not all fires are arson, but some are.

All of this of course feeds onwards into another categorisation of causal factors: mechanical necessity, chance/random factors, intelligently directed contingency. But, that discussion is for another day, save, that we can lay out an explanatory factor flowchart:

. . . and may ponder competing explanations through abductive inference to best explanation so far:

But, are all entities contingent? No. For, as numbers already indicate, some entities are necessary for any world to exist, as they are inherently part of the framework. To see that, we call on the resources of the principle of distinct identity. Notice, above, we contrasted W1 and W2 etc, drawing out differences even between neighbouring possible worlds. Thus, each world Wk has its own distinct identity and we may freely identify some A such that Wk = (A|~A}. Already we see by partition, twoness. From twoness we infer to one-ness and nullity as emptiness {}. Then, following von Neumann:

{} –> 0

{0} –> 1

{0,1} –> 2

. . .

[Thence, the onward panoply of the transfinites.]

Etc.

Numbers, thus the logic of structure and quantity — the substance of Mathematics, are inherent to the framework of any possible world. This already implies reality in some sense of abstract being. And it further seems that the abstract model worlds of core Mathematics can be sub-worlds integrated with physical ones. Peculiar indeed.

Numbers, we have seen, are necessary beings.

Such NB’s (as opposed to contingent ones, CB’s) have peculiar properties. For one, no Wk is possible in which NB’s do not exist. Try to ponder a world where 2-ness does not exist, or begins to exist or ceases from being. Impossible. Thus, we see a constraint that highlights worlds that are impossible of being. And, a serious candidate NB will therefore be either impossible of being (square circle logic) or it will be present in any actualised Wk. One implication is, that a world is — manifestly so, implies that all truly necessary beings eternally are.

Where of course, the God of Ethical Theism is a serious candidate necessary being. He is either impossible of being or actual. And yes those who claim to know or to practically know he does not exist in actuality have taken up a huge burden of proof. One, I daresay, has never been met. (If you doubt, just fill in one of those pesky UD blanks: ______ Where, for example, post Plantinga, the problem of evil is effectively dead.)

All of this points onward to a further thorny issue: intelligibility (at least to God! and partly, to us!) of the world.

That is, some form or other of the principle of sufficient reason. I here put up a weak form sufficient to pursue logic of being:

[PSR, weak (investigatory) form:] Of any particular thing A that is

[. . . or (ii) is possible, or even (iii) is impossible],

we may ask, why it is

[. . . or (ii’) why it is possible, or (iii’) why it is impossible],

and we may expect — or at least hope — to find a reasonable answer.

Of course, for any given case, X, we may simply directly proceed to ask why is X so, or why is X possible or why is X impossible, and seek a reasonable answer. So, the weak form as it stands is unobjectionable.  It is also central to science and to the project of being rational, responsible creatures.

Linked, we may ponder a claim some have advanced, that our world is just one stage in an ongoing infinite in the past and potentially infinite in the future, chain of successive, causally-temporally linked stages, back to the singularity and beyond without limit:

. . . s-n, s- (n-1), . . . s-2, s-1, NOW, s+1, s+2 . . .

This has been repeatedly debated here at UD, and I continue to hold that the proposed past infinity is incoherent as it has no basis to arrive at now in finite successive stages. Others have claimed differently. I doubt such claims as actual stepwise succession in finite steps cannot ever span a transfinite range.

So, we here see causality and intelligibility of being as key first principles of rationality. END

PS: As issues on dynamics and trajectories have been raised, here is a basic system diagram:

This gives a context to consider initial state, traversal across phase space and impact of intervening forces, noise etc.

112 Replies to “Logic & First Principles, 4: The logic of being, causality and science”

1. 1
kairosfocus says:

Logic & First Principles, 4: The logic of being, causality and science

2. 2
PaoloV says:

These topics you explain in this series should be included in a required course in the educational system.

3. 3
kairosfocus says:

PaV, one would only hope so. First principles of reason should be well known by every high school kid, much less undergrad. If that were so, much of the ruinous rhetoric of our day would fall flat. KF

PS: Some of the stuff would be hard to find, though. It’s not just summarised from readings or prior debates at UD or elsewhere.

4. 4
Pater Kimbridge says:

From the OP:

Numbers, thus the logic of structure and quantity — the substance of Mathematics, is inherent to the framework of any possible world.

One must be careful not to slide into a pit of equivocation here. The universe has structure and quantity, but numbers are an invention of Man. In fact, all of mathematics is an invention of Man.

It bothers me when people refer to the “mathematical universe”. There is no math in the universe. To believe that there is, is a mind-projection fallacy.

5. 5
ET says:

Pater K:

It bothers me when people refer to the “mathematical universe”.

We live in a mathematical universe

There is no math in the universe.

6. 6
Belfast says:

PK is correct.
Nature knows nothing of anything bigger than 1. Two atoms are different from each other in location at least.
Mathematics invented by man enables man to quantify order and states and understand patterns at various levels.

7. 7
ET says:

The universe couldn’t have been intelligently designed without mathematics.

8. 8
kairosfocus says:

PK & B:

We are first not just dealing with the physical cosmos when we speak of reality or more particularly of possible worlds.

Next, the principle of identity is integral to there being a distinct world. Once that obtains we have Wk = {A|~A} thus distinction and twoness — as is exemplified by two distinct atoms. From which follow the whole panoply of numbers.

We discover or recognise this, we do not invent it and project it unto an otherwise unknowable noumenal world of things in themselves, unreachable across an ugly gulch. That is, we see here the pernicious effects of one of those little errors in the beginning. (You may wish to read the part of #3 that speaks to that.)

In a nutshell, Bradley corrected Kant’s error long since by showing how it is self-referentially incoherent.

Reality is in part intelligible, or even knowable and this includes the logic of structure and quantity.

Yes, we may go beyond the core and create abstract model entities or worlds which may be very different from possible worlds, e.g. a proposed square circle. Such is impossible of being.

But that has nothing to do with the core we discuss which is framework to reality as you actually recognise.

KF

9. 9
kairosfocus says:

ET, an ordered system of reality with quantitative aspects already embeds mathematics in its framework. This begins with the abstract identity of a distinct possible world Wk as different from others. KF

10. 10
ET says:

In my opinion mathematics permeates the universe and was discovered by us. And also in my opinion, Srinivasa Ramanujan is a prime example of just that.

11. 11
Ed George says:

For what it’s worth, I have to side with PK on this.

12. 12
ET says:

For what it is worth- I would love to see PK or anyone who sides with PK actually make their case.

13. 13
kairosfocus says:

EG, make your case: _______ KF

14. 14
kairosfocus says:

PS: F H Bradley in response to the Kantians:

We may agree, perhaps, to understand by metaphysics an attempt to know reality as against mere appearance, or the study of first principles or ultimate truths, or again the effort to comprehend the universe, not simply piecemeal or by fragments, but somehow as a whole [–> i.e. the focus of Metaphysics is critical studies of worldviews] . . . .

The man who is ready to prove that metaphysical knowledge is wholly impossible . . . himself has, perhaps unknowingly, entered the arena . . . To say the reality is such that our knowledge cannot reach it, is a claim to know reality ; to urge that our knowledge is of a kind which must fail to transcend appearance, itself implies that transcendence. For, if we had no idea of a beyond, we should assuredly not know how to talk about failure or success. And the test, by which we distinguish them, must obviously be some acquaintance with the nature of the goal. Nay, the would-be sceptic, who presses on us the contradictions of our thoughts, himself asserts dogmatically. For these contradictions might be ultimate and absolute truth, if the nature of the reality were not known to be otherwise . . . [such] objections . . . are themselves, however unwillingly, metaphysical views, and . . . a little acquaintance with the subject commonly serves to dispel [them]. [Appearance and Reality, 2nd Edn, 1897 (1916 printing), pp. 1 – 2; INTRODUCTION. At Web Archive.]

15. 15
kairosfocus says:

PPS: Clipping OP:

are all entities contingent? No. For, as numbers already indicate, some entities are necessary for any world to exist, as they are inherently part of the framework. To see that, we call on the resources of the principle of distinct identity. Notice, above, we contrasted W1 and W2 etc, drawing out differences even between neighbouring possible worlds. Thus, each world Wk has its own distinct identity and we may freely identify some A such that Wk = (A|~A}. Already we see by partition, twoness. From twoness we infer to one-ness and nullity as emptiness {}. Then, following von Neumann:

{} –> 0

{0} –> 1

{0,1} –> 2

. . .

[Thence, the onward panoply of the transfinites.]

Etc.

Numbers, thus the logic of structure and quantity — the substance of Mathematics, is inherent to the framework of any possible world. This already implies reality in some sense of abstract being. And it further seems that the abstract model worlds of core Mathematics can be sub-worlds integrated with physical ones. Peculiar indeed.

Numbers, we have seen, are necessary beings.

Such NB’s (as opposed to contingent ones, CB’s) have peculiar properties. For one, no Wk is possible in which NB’s do not exist. Try to ponder a world where 2-ness does not exist, or begins to exist or ceases from being. Impossible. Thus, we see a constraint that highlights worlds that are impossible of being. And, a serious candidate NB will therefore be either impossible of being (square circle logic) or it will be present in any actualised Wk. One implication is, that a world is — manifestly so, implies that all truly necessary beings eternally are.

Kindly show us a world in which distinct identity does not obtain, or two-ness, or one where these are invented or have gone out of being: ________

16. 16
ET says:

Mathematics is the universal language because:

A) Humans invented it

B) It permeates the universe as part of its intelligent design. We merely discovered and actively use it.

17. 17
Ed George says:

KF

EG, make your case: _______ KF

It is my opinion that mathematics is a human invention that can be used to model the world that we see around us. For example e = mc^2 means absolutely nothing without first defining energy, mass and the speed of light.

18. 18
ET says:

It is my opinion that mathematics is a discovery made by humans that can be used to describe and understand the world that we see around us. And that is because the universe was intelligently designed using mathematics.

For example e = mc^2 means absolutely nothing without first defining energy, mass and the speed of light.

True. Nature did not produce those symbols. Humans did invent those and humans also defined them. But there is no reason that such a simple formula should be able to describe the energy and mass relationship other than the universe was intelligently designed using mathematics. And we just need to keep discovering all of its details.

19. 19
john_a_designer says:

So what is EG’s argument?

Either X or Y could be true
EG believes Y
Therefore, Y is true.

In other words, Ed George believes it. That settles it.

20. 20
kairosfocus says:

EG,

Notice where I started — the natural, counting numbers? Rooted in distinct identity and contrast so we see oneness and two-ness again contrasted with nothingness and with onward distinct increments?

That is a natural structure, founded on what is required for a distinct world to exist, distinct identity.

We may extend this in various ways, showing onward quantities and structured patterns that are likewise embedded in any possible world as they are rooted in distinct identity.

You started, not with Math (much less its core) but with an expression in Physics. One, that emerged from pondering the electrodynamics of moving bodies. What this reflects is that we naturally observe quantitative aspects of the world in physical investigations, and that these then naturally find mathematical expression.

Yes, we have defined particular symbols and ways of expressing quantities, e.g. they once thought of vis viva, twice the value of what we know now as kinetic energy. If we had continued with those particular framings, we would have somewhat different equations but the same quantitative, substantial substance.

That is, we find that mathematical frameworks are deeply embedded in physical reality. Indeed the equation you name then explained binding energy mass deficits in observed nuclei of atoms, which then led to nuclear power and weapons.

Let me pause a moment and note an expression in physics, derived from Galileo’s kinematics of uniformly accelerated motion, which is a bit of algebra or else of drawing graphs of such motion and drawing out relationships:

v^2 = u^2 + 2ax

Now, bring to bear Newton’s principle F = ma, and the understanding that work W = Fx

Multiply through by m:

mv^2 = mu^2 + 2max

or,

mv^2 – mu^2 = 2max

i.e.

mv^2 – mu^2 = 2Fx = 2W

Then, divide through by 2:

1/2(mv^2 – mu^2) = Fx = W

Meaning, work done to uniformly accelerate a massive body is the change in its kinetic energy.

This can be extended to the non-uniform case, and will hold.

Much can be built on this.

So, we see here that we are dealing with a tight integration between observation and mathematics, with mutual illumination.

So, physics also points to the natural embedding of the logic of structure and quantity in the world.

KF

21. 21
Ed George says:

In other words, Ed George believes it. That settles it.

That is all any of us can do. Mathematics either exists independent of humans or it is something invented by humans to model our observations. My opinion is that it is the latter. ET and KF believe it is the former. But, unfortunately, there is no way of determining which is true. And, frankly, does it matter?

22. 22
Pater Kimbridge says:

We can’t let language mess with our heads.

We discover regularity, order, patterns, and symmetry in nature

We want to describe it so we can perhaps understand it further and make predictions about it.

So… we invent a language with symbols and operations to describe the elements we observe. We call this math.

By manipulating those symbols in a logical way, we are able to deal abstractly with the phenomena we see in nature.

By combining the observed and the invented into one word “mathematics”, we run the risk of the created thing (math) and the observed thing (patterns and symmetry) being confused.

Of course, if your intention is to fool yourself into seeing design where it may not exist, then this confusion may be just what the doctor ordered.

23. 23
kairosfocus says:

EG,

nope.

What you are doing is losing objective truth and objective warrant.

You live in a world where you know the difference between food and poison. You distinguish between a car coming down the road fast and a street safe to cross. You know a bright red ball on a table different from the rest of the world.

Let that Ball be A, so W = {A|~A}. As the earlier articles (esp. 2 and 3) brought out, A is itself i/l/o distinct characteristics, and contrasts with NOT-A, ~A.

No x in W will be A AND ~A.

Any x in W will be A or else — X-OR — ~A.

This also shows us oneness contrasted with twoness, and indicates nothingness (No x in W . . .), then also invites onward contrasts with 3-ness etc.

The natural numbers are called that for a reason.

And, this is a case where we can have warranted, certain knowledge about the real world and indeed any possible world.

It so happens that we are here exploring the core of the logic of structure and quantity, i.e. Mathematics.

This is not mere opinions and beliefs irrelevant to warrant, it is warranted, credible belief, i.e. objective knowledge.

It truly describes reality and we can show why disbelieving it ends in absurdity.

It is discovered, not invented, just as we readily see that || + ||| –> |||||. Or, in symbols we are familiar with: 2 + 3 = 5 or even II + III = V. Where the latter is actually a short form for the V formed by spreading out our fingers. And yes the X is two such hands back to back.

Just, those symbols rapidly become too cumbersome to work with, e.g. 500 years ago, the long division algorithm using Roman numerals was university grade work. Place value decimal notation long since superseded it and brought that down to grade 3 is it.

KF

24. 24
kairosfocus says:

PK,

nope.

Start again with that bright red ball on a table and the rest of the world around it (review no’s 2 and 3 . . . isn’t it interesting that this exchange on already covered ground appears where that is simply linked), then extend to W = {A|~A} and its corollaries.

The natural numbers are natural, manifest, massivele evident phenomena. That we find ways to symbolise accurately and conveniently then explore the logic of structure and quantity . . . see, a natural sense of what Mathematics is (logic and substantial reality before labels) . . . does not lock this up into an inner, phenomenal world severed from the noumenal world of things in themselves by an impassible ugly gulch.

Notice, F H Bradley’s corrective to the Kantians as was already cited!

Let me do so again:

We may agree, perhaps, to understand by metaphysics an attempt to know reality as against mere appearance, or the study of first principles or ultimate truths, or again the effort to comprehend the universe, not simply piecemeal or by fragments, but somehow as a whole [–> i.e. the focus of Metaphysics is critical studies of worldviews] . . . .

The man who is ready to prove that metaphysical knowledge is wholly impossible . . . himself has, perhaps unknowingly, entered the arena . . . To say the reality is such that our knowledge cannot reach it, is a claim to know reality ; to urge that our knowledge is of a kind which must fail to transcend appearance, itself implies that transcendence. For, if we had no idea of a beyond, we should assuredly not know how to talk about failure or success. And the test, by which we distinguish them, must obviously be some acquaintance with the nature of the goal. Nay, the would-be sceptic, who presses on us the contradictions of our thoughts, himself asserts dogmatically. For these contradictions might be ultimate and absolute truth, if the nature of the reality were not known to be otherwise . . . [such] objections . . . are themselves, however unwillingly, metaphysical views, and . . . a little acquaintance with the subject commonly serves to dispel [them]. [Appearance and Reality, 2nd Edn, 1897 (1916 printing), pp. 1 – 2; INTRODUCTION. At Web Archive.]

KF

25. 25
kairosfocus says:

Folks,

do we begin to see just how deep the rot is in our civilisation?

Let’s go to something more direct again, a self-evident quantitative truth:

|| + ||| –> |||||. Or, in symbols we are familiar with: 2 + 3 = 5 or even II + III = V. Where the latter is actually a short form for the V formed by spreading out our fingers.

Can we accept that this is a natural phenomenon structured into say the realities of our fingers and toes? KF

26. 26
Ed George says:

KF@23, let’s just agree to disagree about what mathematics is and its origin.

27. 27
kairosfocus says:

EG,

the objectivity of core mathematics starting with the naturals or say basic self evident relationships such as ||| + || –> ||||| is not in question: objectivity meaning independent of any particular individual’s perceptions, opinions or views, subject to open warrant on evidence and/or reasoned argument. It is not a matter of seeming true to you but not to me and that’s where it ends.

What is in question, sadly (and with all due respect to a LOT of people caught up in the tides of our times), is willingness to acknowledge that objectivity.

I disagree — for reasonable, well-warranted cause — that rejecting or dismissing a demonstrably objective or self-evident truth is a sign of superior adherence to “tolerance” or the like. Instead, i/l/o known massive trends and agendas, it is a mark of indoctrination to reject objectivity.

That’s not your fault in the first instance, you have been clearly subjected to a lot of conditioning that leads you to react to presentation of objective truth-claims. Likely you have been taught or influenced to perceive that insistence on such truths is intolerant and perhaps a sign of “true believer” or “fascist” tendencies and ever so much more.

Which is a reflection of deliberate marginalisation, stereotyping and scapegoating . . . exactly the opposite of genuine tolerance.

And that, in turn is a sobering warning and sign of where we are now as a civilisation.

My deep, longstanding concern, is that we must take urgent steps to pull back from the brink.

KF

28. 28
kairosfocus says:

PS: Just to illustrate, let’s note a cluster of Dictionaries:

mathematics
math·e·mat·ics (m?th??-m?t??ks)
n. (used with a sing. verb)
The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols.
[From Middle English mathematik, from Old French mathematique, from Latin math?matica, from Greek math?matik? (tekhn?), mathematical (science), feminine of math?matikos, mathematical; see mathematical.]

mathematics (?mæ???mæt?ks; ?mæ??mæt-)
n
1. (Mathematics) (functioning as singular) a group of related sciences, including algebra, geometry, and calculus, concerned with the study of number, quantity, shape, and space and their interrelationships by using a specialized notation
2. (Mathematics) (functioning as singular or plural) mathematical operations and processes involved in the solution of a problem or study of some scientific field
[C14: mathematik (n), via Latin from Greek (adj), from math?ma a science, math?matikos (adj); related to manthanein to learn]
Collins English Dictionary – Complete and Unabridged, 12th Edition 2014 © HarperCollins Publishers 1991, 1994, 1998, 2000, 2003, 2006, 2007, 2009, 2011, 2014

math•e•mat•ics (?mæ? ??mæt ?ks)

n.
1. (used with a sing. v.) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically.
2. (used with a sing. or pl. v.) mathematical procedures, operations, or properties.
[1350–1400; < Latin < Greek math?matik? (téchn?) scientific (craft) =math?mat- lesson, learning + -ik?, -ic; see -ics]

math·e·mat·ics (m?th??-m?t??ks)
The study of the measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are branches of mathematics.

See why my modded form of the definition taught long ago by my first Uni prof in my first Math course at that level is apt: the [study of the] logic of structure and quantity. I didn’t add “space” as that is a form of structure, sets are structures, relationships are structures again and logic needs to be highlighted in this day and age. Quantity is so primary that that form of structure needs to be emphasised.

BTW, on a similar matter, in Newton’s Laws of Motion the first is a special case of the second but is conceptually vital for understanding inertia as well as practically important for statics.

29. 29
Ed George says:

KF@27&28, as I said, we agree to disagree on what mathematics is and where it originates.

30. 30
kairosfocus says:

EG,Kindly explain to us why you disagree with the consensus voice of the dictionaries as cited: _____ KF

PS: Here is Enc Brit online: Mathematics, the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable adjunct to the physical sciences and technology, and in more recent times it has assumed a similar role in the quantitative aspects of the life sciences. [Caution, “science” here is not used in the conventional sense! “Study” would have been a better term, especially if coupled to logic.]

PPS: The Cambridge Dictionary is nice: the study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organizing them

31. 31
Ed George says:

KF@30, all I disagree with is that mathematics is inherent in the universe and not a tool developed by man to model this universe. Show me another intelligence that has developed the same mathematics we use and I might change my opinion.

32. 32
kairosfocus says:

EG, do you disagree that any possible world, Wk will have a distinct identity that marks it as different from any other world, Wj? If so, you have the identity of indiscernibles to deal with. If not, then Wk = {A|~A} inherently obtains and with it nullity, unity and duality. From this we have the inevitability of number as the von Neumann construction elaborates as part of the framework of any world. Whether some other race or civilisation resorts to decimal numbers and defining pi vs using 2pi, or any of thousands of other things first does not change the naturals. Second, using differing numerals for the same quantities does not alter their structural properties and relationships. Third, as I took time to show using kinetic energy, the extension to the physical world goes beyond artifacts such as whether we use vis viva instead or even if we were to use F = dp/dt or Newton’s dot notation, dot-p, or some other form that say the Kzinti might develop, we would simply need a few more steps to get back to the same physics. KF

33. 33
Ed George says:

KF@32, yes, we are capable of using mathematics to describe the physical world. Nobody is disputing that. But is that because humans are capable of devising a system to do so, or because the system exists independent of humans? I think it is the former, you think the latter. But the one thing that is certain is that we don’t have the tools necessary to discern which is true. Until then, we will just have to agree to disagree. I am fine with that because, ultimately, who cares? Does it make any difference to our lives?

34. 34
ET says:

Mathematics is universal- not just here on earth but also with every other technologically capable civilization out there.

On Earth we had Srinivasa Ramanujan who discovered many formulae is his all too brief life.

If you accept a hunter/ gatherer society then you can see that mathematics would just fall right out of such a scenario. Basic math would be a way of life.

35. 35
Nonlin.org says:

Actually, “all bachelors are unmarried” fails as some bachelors of science are married.

Necessity defined as “true in all possible worlds” is at best unknowable. We will never know “all possible worlds”. Experience shows that “true in all possible worlds” has been proven false way too many times.

36. 36
kairosfocus says:

Nonlin, There is a very specific point on the table. To have some world Wk as opposed to any other world Wj, there has to be a distinction. Or else, they are identical (which makes them indiscernible). In effect two labels for the same thing much as evening and morning star turned out to be the same planet, known as Venus. No, we simply do not need to enumerate all worlds or observe them. Once distinction obtains, we have W = {A|~A}, thus duality, unity and nullity. These are not empirical issues, they turn on what is needed to have distinct identity. Indeed, the concept of experiences demonstrating a result pivots on distinct identity and its correlates, non-contradiction and excluded middle. You are in the position of relying on what you would set aside. With 0,1,2 in play, 3 and onward come in. Number is embedded in the framework for any distinct world to exist. Other things extend from that. KF

PS: One Governor Plum later, to claim that properties across all possible worlds are unknowable is to imply knowing something about all possible worlds.

PPS: The objection on bachelors turns on an equivocation. Unmarried male is the obvious context of relevance.

37. 37
kairosfocus says:

EG, if you scroll up you will see that I clearly distinguished what is universal from what is culturally conditioned. Cf. 20, 23, 25. For example, 20: “Yes, we have defined particular symbols and ways of expressing quantities, e.g. they once thought of vis viva, twice the value of what we know now as kinetic energy. If we had continued with those particular framings, we would have somewhat different equations but the same quantitative, substantial substance.” In 23, by contrast, numbers and self-evident relationships: “The natural numbers are called that for a reason . . . . It so happens that we are here exploring the core of the logic of structure and quantity, i.e. Mathematics. This is not mere opinions and beliefs irrelevant to warrant, it is warranted, credible belief, i.e. objective knowledge . . . . It is discovered, not invented, just as we readily see that || + ||| –> |||||. Or, in symbols we are familiar with: 2 + 3 = 5 or even II + III = V. Where the latter is actually a short form for the V formed by spreading out our fingers.” That we happen to have five fingers only makes this more convenient for us, pebbles (likely to occur on any terrestrial planet) will make the same point. KF

38. 38
kairosfocus says:

Folks, ironically this should have happened a couple of threads back. This one actually highlights causality and sufficient reason! KF

39. 39
Ed George says:

KF@37, does a stone falling from a cliff fall at a rate determined by mathematics or do the mathematics describe the rate at which the stone falls?

40. 40
john_a_designer says:

Here are some questions for Ed George:

In your opinion which of the following numbers is a prime number?

18397497

18397499

18397501

Is it possible that none of them are prime numbers?

Is it possible that all of them are primes?

Does human belief or opinion have anything to do with whether or not a number is prime?

Were prime numbers invented or discovered?

41. 41
kairosfocus says:

EG,

This needs a two-parter. First, the underlying issues.

Do you notice, that you begin from a set of alphanumerical characters, each distinct (and in a digital world, each expresses through a numerical bit-string, usually based on the 7-bit ASCII code or the like)?

In short, just to object, you are forced to rely on the power of the principle of distinct identity.

It is that pervasive and it is that powerful.

Where, much more broadly, this reflects the nature of language, a key tool of thought. Just to have complex thoughts sufficient to explore mathematical realities we must have language capable of handling abstractions.

That language, as is notorious, finds expression in a considerable array of meaningful symbols. Which of course may and do vary within human discourse on Math, e.g. there were things comparable to but not identical with our modern trig functions in certain ancient work. Similarly ponder more specialised trig functions such as the haversine used in navigation — observe how it can be used whole or reduced to combinations of other functions, indeed sine, cos and exponential can be reduced to power series form. There is no reason to imagine that Vulcans, Kzinti, Tree-cats etc doing mathematics will not use linguistic elements and functional forms that are at least related to what we use.

Now, many of us will have been conditioned to leap from linguistic elements to language is inherently cultural thence that we here deal with culturally relative institutions and features. Game over!

NOT.

For, again and again, just to use such symbols we have been forced to rely at every stage on their distinct identity.

My point in the OP, in this thread so far and elsewhere therefore is that this principle instantly implies distinction, A vs ~A, thus two-ness.

Twoness or duality entails one-ness or unity. In turn that points to nullity, as can be readily seen from von Neumann’s construction. Instantly, we are at the threshold of the key mathematical structure, the set of natural counting numbers.

What happens when you partition your fingers into a two-set and a three-set of distinct elements then observe that their join or union yields a five-set now shows a quantitative relationship, one that is self-evident:

|| + ||| –> |||||

Culturally, using standard Indo-Arabic symbols as modified by the West (well do I recall my Indian profs and how the way they consistently, unconsciously wrote 0 showed their distinct cultural background):

2 + 3 = 5

Notice, too, how the pipe characters above are using distinct location as a marker of distinct identity.

Similarly, in 40 above JAD points to a property of a key subset of the Naturals, the primes.

That property of not being evenly split up into factors less than themselves and distinct from unity inheres to their distinct nature as particular numbers. Where, in the particular case above, it so happens that I routinely use a relationship between the first three primes: 2, 3, 5.

Again, that distinct property that is so useful in Mathematics and in crypography is not a cultural invention, it is a discovery, as is the once shocking revelation that the primes — a proper subset of the naturals that by their nature must become ever sparser as we go down the chain of naturals — are endless in number.

Let’s turn to something else that we now seldom study as an axiomatic system and separate subject in schools in its own right that reveals ever so many structural and quantitative properties of space[other than to highlight the famed fifth postulate], Geometry. For, how we now look at Euclidean Geometry may be crucial to how we have been led to imagine that Mathematics is an invention rather than an exploration.

Consider the relationships, 3-4-5, 12-5-13, 1-1-sqrt-2 and 1-2-sqrt-3. They are of course special forms of the Pythagorean theorem. There are many others, revealing properties of triangles, squares, lines, circles, ellipses and other figures, including previews of what the calculus would draw out.

The last two we often find in geometry sets of instruments and in drafting instruments. Set squares, we call them. I bought a set as recently as a few months back.

The first, appears in archaeology by way of exploiting yet another self-evident and very useful readily discoverable truth: 3 + 4 + 5 = 12, so builders of old would braid or twist a rope and tie in it evenly spaced knots making twelve segments. This is also fairly close to being sides of a square so that the limbs will not be disproportionate.

Stretch the rope into a triangle, level it properly and you have a guaranteed right angle. (Modern builders may lay out a similar object using 6-foot, 8-foot and 10-foot timbers.)

To assure squareness of a room or another object, measure diagonals for equality.

And so forth.

Where of course the half-square triangle shattered the cultural expectations of the Pythagoreans by exposing the utterly unexpected property: irrational numbers.

Were these invented?

Nope, they were discovered and they were eventually integrated into a coherent system, Geometry.

But, doesn’t the Fifth, parallel-lines or angle sum triangle postulate demonstrate that all of this is an imposed artificial system?

No.

The cases of practical plane geometry that come out through trying to build physical structures give a clue.

I often use the construction of the Argand plane from the reals (by way of orthogonal rotation) to set up x and y axes intersecting at the origin, o [not zero]. Now, ponder the equation of the straight line, generic form, y = m*x + c. Now, consider a case where m is some negative, fixed value, say -q. y = -q*x + c. Then, produce a set of positive values [thus y-axis intercepts] for c, c1, c2, c3 . . .

Plot the resulting lines on the plane. We will see a set of nested similar triangles between y and x axes and of course parallel lines that run at the slope -q. By the force of algebra, those lines will never converge for any particular value of x and values cj and ck that we may pose or even symbolise (as I just did).

The 5th postulate identifies a property of planar space in two and three dimensions that is real.

What the C19 Mathematicians discovered is that such a space is not the only possible or relevant one. And indeed, in the large, our physical cosmos — per Relativity — follows Riemannian properties. But already, sailing across Earth established that the surface in the large was spherical not planar and the geometry of perspective shows similar possibilities.

So, no, non-Euclidean Geometry does not turn Mathematics into a cultural process of invention that is essentially subjective and relative. There are culture-relative aspects, but those are constrained by objective structural and quantitative features tied to the pervasive presence and power of distinct identity.

The rise of such Geometries then brought out a property you have indeed recognised: axiomatic systems allow us to form abstract, logic model possible worlds.

The phrase “possible worlds” should give a clue.

For, we then see that necessary being, framework properties that crop up in those possible worlds will perforce extend to our actual one. By virtue of the nature of necessary entities that are framework for any world to be. Of course, distinct identity thus two-ness and the system of natural numbers is a key case in point.

The extension to rationals and the continuum will be equally relevant and these too will be found in any development of mathematics by Kzinti etc of sufficient complexity to be relevant. I am fairly sure that any culture sufficiently sophisticated to make threads and ropes or the like (strips of hide will do) will discover properties of the continuum.

Abstract reflection will lead to a development of Mathematics, and that will have cultural properties, yes; but must respond to objective ones also, starting with the naturals.

KF

42. 42
kairosfocus says:

EG,

Now, secondly, your example from physics:

does a stone falling [–> trajectory in a space] from a cliff fall at a rate [–> an abstract entity, expressed as a ratio and indeed pointing to infinitesimals and the calculus] determined by mathematics or do the mathematics describe the rate at which the stone falls?

Do you see how abstract, objective, distinct things tied to the logic of structure and quantity are inescapably embedded in your example?

The stone falls [oops, we observe or perceive it falling — is that itself objective?], describing a trajectory in space and time [h’mm, how do we define and use units of length and time?]; leading to issues of displacement, velocity, acceleration and (if the height is enough) questions of retardation. Onward lie matters of fields of influence; here, gravitational, thence the fabric of spacetime.

Next, you have put cart before horse.

Falling presupposes space, length, duration, succession, time and trajectory, which are inherently Mathematical, as we already saw. So soon as Kzinti can twist up or braid a thread or rope, they are dealing with these things. All that remains is to ponder and structure in a rational way how they/we ponder. The study of the logic [= rational framework] of structure and quantity comes out.

Rate is even more Mathematical and abstract: dx/dt –> v, dv/dt –> a, da/dt –> “jerk,” with F = dP/dt lurking. (Nb: here is useful, on a railroad it is helpful to do a smooth lead-in to a curve so that passengers are not jolted . . . cf here: “Railroad tracks and roads ( not always ) are designed to enter from a straight section into a curve with a transition called a clothoid where the radial acceleration eases from 0 to v^2/R, in which case R is a variable from the entrance into the curve to the value at the circular curve . . . . It is applied, or should be, for other transitions in curvature for machines, roller coasters in certain critical curve transitions, aircraft maneuvers.”) To ask the question you have had to rely on framework properties of reality that are present in any possible world: numbers, continua, infinitesimals, limits, sequences and series etc.

So, the attempt to wedge apart the abstract mathematical properties and the physical processes fails so soon as we understand that we are making inherently mathematically involved observations and are seeking a mathematical form that describes then explains what is happening.

The math and the physics are inextricably intertwined and entangled.

The challenge is ill-formed.

KF

43. 43
kairosfocus says:

Let us for now leave the exercise of feeding in the numbers to EG.

Then, let us see if he is able to understand the objectivity of that very abstract property of being or not being prime; of an already abstract but very real entity, a natural number.

BTW, just to help him — our day is conceptually deeply impoverished and dumbed down — has he ever seen or touched woman, man, love, child, truth, rationality? (That is, the real world is full of abstract entities that capture in common aspects of particular things. This points onward to the problem of the one and the many.)

KF

44. 44
ET says:

Ed George:

does a stone falling from a cliff fall at a rate determined by mathematics or do the mathematics describe the rate at which the stone falls?

Both. The stone falls at a rate determined by the mathematics of the universe and that is why the mathematics describes the rate at which the stone falls.

45. 45
Pater Kimbridge says:

Just when I think you are agreeing with myself and Ed George, you seem to circle back to the same confusion:

KF:
Falling presupposes space, length, duration, succession, time and trajectory, which are inherently Mathematical.

That’s like saying love, honor, and duty are inherently Linguistic because we have words to describe them.

46. 46
Pater Kimbridge says:

ET:
The stone falls at a rate determined by the mathematics of the universe and that is why the mathematics describes the rate at which the stone falls.

The stone falls at a rate determined by the properties of the universe and that is why mathematics can be used to describe the rate at which the stone falls.

Fixed it for you.

47. 47
ET says:

The properties of the universe are defined by its mathematics and that is why mathematics can be used to describe the rate at which the stone falls.

Fixed it for YOU.

48. 48
ET says:

Peter K:

That’s like saying love, honor, and duty are inherently Linguistic because we have words to describe them.

No, that isn’t even close.

Try again. Or forget it.

49. 49
kairosfocus says:

PK,

First, have you done calculus or at least high school physics? That may help me understand the communication gaps here. (I notice we are running 1 – 2 threads late on issues, this is about causality and intelligibility of being.)

A falling object is translocating across space, which is to all intents a continuum at relevant scale. That brings to bear the real numbers, three sets beyond the natural counting numbers.

In effect, allow me to define negative numbers so that [-q] is additive inverse of q, summing to zero. Rationals then take ratios of naturals. To get continuum you have to bring in splitting of numbers to whole and fractional parts, then allow for infinite chains of summed fractions, converging to the required value such as sqrt-2. In the continuum, between ANY two numbers an intervening real value can be defined. That is, there is no distinct last valid nearest neighbour to L or RH of any specific value. That’s tied to why the continuum is more intensely transfinite than the counting numbers.

All of these pivot on distinct identity.

Now, rates of fall have to do with displacement x changing over time [another continuum] so we get velocity v = dx/dt, dx and dt infinitesimals (next to zero values so that powers go very fast to effective zero . . . long story] on the non-standard analysis approach. On standard analysis one, we look at sequences and series and limits. Beyond, acceleration is dv/dt, which is tied to force as rate of change of momentum, F = dP/dt, in constant mass case F = ma.

Looking at the space side, increment in work by falling under relevant forces is dW = F.dx, which in vacuo would go into increased kinetic energy (already discussed). In air, displacement of air, vibrations, potentially even heating are involved. In turn, heat is tied to the molecular behaviour, which involves statistics and ultimately quantum issues and statistical thermodynamics. All, deeply involved with mathematical aspects of reality.

All of this thus inextricably intertwines mathematical properties of space, time and bodies into the physics. All of which bring in the logic of structure and quantity and involve distinct identity. Once one takes more than a superficial look.

Repeat, the question is ill-posed.

The objectivity of a falling rock is deeply involved with the logic of structure and quantity.

KF

PS: If we cannot get a falling rock straight, the far more involved concepts love, honour, duty etc will be intractable. Let’s just note, nominalism (it’s just a label for cases arbitrarily collected on perceptions inherently unable to cross the ugly gulch to things in themselves) is dubious. Love, honour and duty are moral and spiritual things that transcend the physical but impact on it. For example rational discourse pivots on known moral duties to truth, right reason, fairness etc, thence moral government. Our presence as reasonable and responsibly free creatures points to moral government at the root of reality.

50. 50
Pater Kimbridge says:

KF:

1) I had two years of calculus on the way to obtaining an engineering degree. Made A’s and B’s

2) I agree with everything in your last post except the phrase “mathematical properties”

3) The universe is the territory. Math is the map

4) You should not confuse the map for the territory.

5) Confusing the territory for the map might lead one to mistakenly think that the territory was designed by a map-maker.

51. 51
Mung says:

There is entirely too much either/or being presented as if there are only two options. Things are not binary Winston.

52. 52
kairosfocus says:

PK, okay, we are on at least a similar wavelength mathematically so the issue is worldviews. In that context, kindly note the difference between the study and particular, culturally bound symbolism and knowledge base for the moment on one hand and the substance on the other. Namely, mathematics as to its essence is the logic — rational framework — of structure and quantity. Therefore, so soon as the world or more narrowly the physical cosmos has quantitative and/or structural features as a distinct ordered system of reality, it has mathematical aspects and properties. In the case of the falling rock, it is moving across space and does so at a rate that itself changes. Already, space and rates are inextricably mathematical; as was just summarised. The physics simply cannot be severed from quantitative and structural features of reality and their order. Yes, there are physical forces at work, tracing to gravity, which is in turn a reflection of the warping of spacetime by a planet-sized, planet-mass [~6 *10^24 kg] object. KF

53. 53
ET says:

Peter K:

3) The universe is the territory. Math is the map

In this case the territory was designed by the map maker. The universe was intelligently designed using mathematics.

54. 54
EricMH says:

If math is made up by man, why doesn’t my system of gnomatics work just as well, where gnomes are magical such that 1 gnome plus 1 gnome equals any number of gnomes you want, except if one of the gnomes is named Sam, in which case 1+1=purple? Am I just being oppressed by the anti-gnomiacs because no one has adopted gnomatics in preference to mathematics? PK et. al., will you go on record supporting me against the status quo in the engineering department?

55. 55
Ed George says:

EricMH

If math is made up by man, why doesn’t my system of gnomatics work just as well,…

If your geomatics accurately modelled observed reality then it would work just as well. Newton’s mathematics works sufficiently well to put people on the moon and probes on Mars because it models reality sufficiently well to do so. However, it does not model reality sufficiently well enough to operate a GPS system.

56. 56
Heartlander says:

The map’s legend/key is designed and applies to the map (universe) in toto as a ‘rational framework’…

57. 57
jawa says:

ET @48&53:

Pater K not Peter K
🙂

58. 58
ET says:

jawa @ 57- It was a mutation- 0110 0001 (ascii for a) was mutated to 0110 0101 (ascii for e).

You see that is how all the different books were written starting from one. 😎

So let’s see- so far I have referred to PeterA as Ed George and now Pater as Peter. I have become my parents who used to go through the list of us before stopping at the right one to scold. And when I was the only one left for some reason my mother had to throw Jesus on that list.

59. 59
john_a_designer says:

Everyone involved in this discussion are missing some obvious points. As Kf pointed out, my point up @ #40 is that numbers have properties and those properties have been discovered not invented. So how can anyone claim that math (or if you’re a brit, “maths”) is a human invention? That’s not to say that human’s didn’t invent the symbolism used to do mathematics. But without the existence of mathematical truth there would be no reason for the symbolism. Furthermore, the development of mathematics preceded the advances of the physical sciences by centuries.

Please notice, I intentionally chose three numbers @#40 for a reason, and the questions I asked are important because they also have something to do with the property of numbers– specifically properties that go beyond the fact whether any of the numbers listed is prime. Does anyone see what I am trying to get at?

60. 60
Heartlander says:

I argue, as others have done before me, that mathematical concepts and ideas exist objectively, outside of the physical world and outside of the world of consciousness. We mathematicians discover them and are able to connect to this hidden reality through our consciousness. If Leo Tolstoy had not lived we would never have known Anna Karenina. There is no reason to believe that another author would have written that same novel. However, if Pythagoras had not lived, someone else would have discovered exactly the same Pythagoras theorem. Moreover, that theorem means the same to us today as it meant to Pythagoras 2,500 years ago.
Edward Frenkel

61. 61
EricMH says:

@EG, aha, so math is not purely made up by man. We are describing a structure in reality. Got it. Not sure what your disagreement is, then.

62. 62
Ed George says:

EricMH

@EG, aha, so math is not purely made up by man. We are describing a structure in reality. Got it. Not sure what your disagreement is, then.

My disagreement, and I think PK’s as well, is minor in nature and really involves two different ways of looking at the same thing. The way that it is presented here implies that mathematics is a causal force, which it obviously isn’t. I think PK’s analogy is appropriate. The map doesn’t come before the landscape, it is a tool used to model the landscape.

A rock falling from the sky in a vacuum will accelerate due to the gravitational attraction of the two masses. We are able to accurately model this using calculus, but calculus doesn’t cause the rock to accelerate.

63. 63
EricMH says:

@EG, I probably missed it, but I have not seen anyone claiming math is causal.

64. 64
kairosfocus says:

EG (& PK): Note, in measuring displacement of the falling rock, we would use an arbitrary unit, the metre (originally based on earth’s circumference). Did that mean that length is an arbitrary, culturally derived property used in hopes of being a useful model for reality? No. Length is an objective property [dimension, L), we just created a reference quantity, the unit. Then, we report a particular case as a ratio, so many times the unit, the metre. Length, of course is a quantitative, structural property found in the world and thus manifests features amenable to the study of the logic of structure and quantity. KF

65. 65
kairosfocus says:

JAD, as you are being studiously ignored, the latter two form a successive pair of primes separated by the intervening even number. The former is divisible by three as can be confirmed by adding its digits: 48. And yes, we discover properties of mathematical nature in those abstract but oh so useful things the natural numbers. KF

66. 66
kairosfocus says:

EMH, mathematical properties constrain what is possible. For instance 2 + 3 will yield 5, not 4 or 6. Investing money yields a result through the earning power and ownership, not magic of numbers, but the numbers are tied to realities. This is why pyramid or Ponzi schemes are disguised theft. Similarly wave properties implied a dot of light in the shadow of a tiny circular dot, which was used to dismiss the wave theory. Then someone tested and there it was: https://en.wikipedia.org/wiki/Arago_spot . KF

67. 67
ET says:

Ed George:

A rock falling from the sky in a vacuum…

Oh my. Talk about a contradictory statement. Or is the rock in a vacuum cleaner?

The point is either you have a sky, meaning there is an atmosphere, or you have no air, no sky and a therefore vacuum.

68. 68
kairosfocus says:

EG, did anyone say calculus — study of rates and accumulations of change — causes a rock to accelerate? No. What has been said is that acceleration is deeply embedded with matters of structure and quantity as was shown, the objection is ill-formed. KF

69. 69
Ed George says:

KF

No. What has been said is that acceleration is deeply embedded with matters of structure and quantity as was shown, the objection is ill-formed. KF

You are certainly entitled to your opinions.

70. 70
Ed George says:

ET

Oh my. Talk about a contradictory statement. Or is the rock in a vacuum cleaner?

The point is either you have a sky, meaning there is an atmosphere, or you have no air, no sky and a therefore vacuum.

A sky is any view looking up from a planetary body. I said “in vacuum” because acceleration due to gravity is affected by atmospheric friction. The math is simpler.

71. 71
ET says:

Ed George:

A sky is any view looking up from a planetary body.

If there isn’t any air then who is looking up? And guess what? The rock will fall at the same rate regardless if anyone is there to calculate it.

It took mathematics to get the laws of nature they way they are. Mathematics permeates everything. There is a mathematical universal structure- to be DISCOVERED.

72. 72
Ed George says:

ET

If there isn’t any air then who is looking up? And guess what? The rock will fall at the same rate regardless if anyone is there to calculate it.

The rate a rock falls will depend on gravity, which is not a constant from one location on earth to another, and on drag from the friction of any atmosphere. In fact, it won’t fall at the same rate in two corners of a small room. But this is really getting off topic.

73. 73
kairosfocus says:

EG, scroll up and see why Kantianism fails. As for what acceleration is, there is no question but that it is a highly mathematical phenomenon, second rate of change of displacement with time. In turn, displacement is itself highly mathematical, tied to space, the continuum and more. Even time is a mathematical, structural and quantitative property of the world. Those are not matters of one opinion vs another, they are objective. KF

74. 74
bornagain77 says:

If math is “just a map” of the territory, it is an awfully peculiar type of map.

I know of no other map that can be infused on top of a material substrate, i.e. on top of the territory, and tell the territory how to behave.

Describing Nature With Math By Peter Tyson – Nov. 2011
Excerpt: Indeed, many of the technologies you and I enjoy every day simply would not work without mathematics. When you do a Google search, you’re relying on 19th-century algebra, on which the search engine’s algorithms are based. When you watch a movie, you may well be seeing mountains and other natural features that, while appearing as real as rock, arise entirely from mathematical models. When you play your iPod, you’re hearing a mathematical recreation of music that is stored digitally; your cell phone does the same in real time.
“When you listen to a mobile phone, you’re not actually hearing the voice of the person speaking,” Devlin told me. “You’re hearing a mathematical recreation of that voice. That voice is reduced to mathematics.”
http://www.pbs.org/wgbh/nova/p.....-math.html

Indeed, the computer that you are sitting in front of right now would not exist were it not for the causal efficacy of the immaterial entities of logic, math, and even thoughts themselves.

Recognising Top-Down Causation – George Ellis
Excerpt: page 5: A: Causal Efficacy of Non Physical entities:
Both the program and the data are non-physical entities, indeed so is all software. A program is not a physical thing you can point to, but by Definition 2 it certainly exists. You can point to a CD or flashdrive where it is stored, but that is not the thing in itself: it is a medium in which it is stored.
The program itself is an abstract entity, shaped by abstract logic. Is the software “nothing but” its realisation through a specific set of stored electronic states in the computer memory banks? No it is not because it is the precise pattern in those states that matters: a higher level relation that is not apparent at the scale of the electrons themselves. It’s a relational thing (and if you get the relations between the symbols wrong, so you have a syntax error, it will all come to a grinding halt). This abstract nature of software is realised in the concept of virtual machines, which occur at every level in the computer hierarchy except the bottom one [17]. But this tower of virtual machines causes physical effects in the real world, for example when a computer controls a robot in an assembly line to create physical artefacts.
Excerpt page 7: The assumption that causation is bottom up only is wrong in biology, in computers, and even in many cases in physics, ,,,
The mind is not a physical entity, but it certainly is causally effective: proof is the existence of the computer on which you are reading this text. It could not exist if it had not been designed and manufactured according to someone’s plans, thereby proving the causal efficacy of thoughts, which like computer programs and data are not physical entities.
http://fqxi.org/data/essay-con.....s_2012.pdf

Supplemental note: Immaterial information is now shown to be a physically real entity that possesses, of all things, a ‘thermodynamic content’. i.e. immaterial information, contrary to materialistic presuppositions, has now been shown to have causal power.

Demonic device converts information to energy – 2010
Excerpt: “This is a beautiful experimental demonstration that information has a thermodynamic content,” says Christopher Jarzynski, a statistical chemist at the University of Maryland in College Park. In 1997, Jarzynski formulated an equation to define the amount of energy that could theoretically be converted from a unit of information2; the work by Sano and his team has now confirmed this equation. “This tells us something new about how the laws of thermodynamics work on the microscopic scale,” says Jarzynski.
http://www.scientificamerican......rts-inform

The preceding work has been extended:

New Scientist astounds: Information is physical – May 13, 2016
Excerpt: Recently came the most startling demonstration yet: a tiny machine powered purely by information, which chilled metal through the power of its knowledge. This seemingly magical device could put us on the road to new, more efficient nanoscale machines, a better understanding of the workings of life, and a more complete picture of perhaps our most fundamental theory of the physical world.
http://www.uncommondescent.com.....-physical/

Information engine operates with nearly perfect efficiency – Lisa Zyga – January 19, 2018
Excerpt: Physicists have experimentally demonstrated an information engine—a device that converts information into work—with an efficiency that exceeds the conventional second law of thermodynamics. Instead, the engine’s efficiency is bounded by a recently proposed generalized second law of thermodynamics, and it is the first information engine to approach this new bound.,,,
The generalized second law of thermodynamics states that the work extracted from an information engine is limited by the sum of two components: the first is the free energy difference between the final and initial states (this is the sole limit placed on conventional engines by the conventional second law), and the other is the amount of available information (this part sets an upper bound on the extra work that can be extracted from information).
To achieve the maximum efficiency set by the generalized second law, the researchers in the new study designed and implemented an information engine made of a particle trapped by light at room temperature. Random thermal fluctuations cause the tiny particle to move slightly due to Brownian motion, and a photodiode tracks the particle’s changing position with a spatial accuracy of 1 nanometer. If the particle moves more than a certain distance away from its starting point in a certain direction, the light trap quickly shifts in the direction of the particle. This process repeats, so that over time the engine transports the particle in a desired direction simply by extracting work from the information it obtains from the system’s random thermal fluctuations (the free energy component here is zero, so it does not contribute to the work extracted).
One of the most important features of this system is its nearly instantaneous feedback response: the trap shifts in just a fraction of a millisecond, giving the particle no time to move further and dissipate energy. As a result, almost none of the energy gained by the shift is lost to heat, but rather nearly all of it is converted into work. By avoiding practically any information loss, the information-to-energy conversion of this process reaches approximately 98.5% of the bound set by the generalized second law. The results lend support for this bound, and illustrate the possibility of extracting the maximum amount of work possible from information.
https://phys.org/news/2018-01-efficiency.html

And in the following article, which has direct bearing on the ‘math is a just a map of the territory’ analogy, the author states, “Fifteen years ago, “we thought of entropy as a property of a thermodynamic system,” he said. “Now in information theory, we wouldn’t say entropy is a property of a system, but a property of an observer who describes a system.”,,,”

The Quantum Thermodynamics Revolution – May 2017
Excerpt: the 19th-century physicist James Clerk Maxwell put it, “The idea of dissipation of energy depends on the extent of our knowledge.”
In recent years, a revolutionary understanding of thermodynamics has emerged that explains this subjectivity using quantum information theory — “a toddler among physical theories,” as del Rio and co-authors put it, that describes the spread of information through quantum systems. Just as thermodynamics initially grew out of trying to improve steam engines, today’s thermodynamicists are mulling over the workings of quantum machines. Shrinking technology — a single-ion engine and three-atom fridge were both experimentally realized for the first time within the past year — is forcing them to extend thermodynamics to the quantum realm, where notions like temperature and work lose their usual meanings, and the classical laws don’t necessarily apply.
They’ve found new, quantum versions of the laws that scale up to the originals. Rewriting the theory from the bottom up has led experts to recast its basic concepts in terms of its subjective nature, and to unravel the deep and often surprising relationship between energy and information — the abstract 1s and 0s by which physical states are distinguished and knowledge is measured.,,,
Renato Renner, a professor at ETH Zurich in Switzerland, described this as a radical shift in perspective. Fifteen years ago, “we thought of entropy as a property of a thermodynamic system,” he said. “Now in information theory, we wouldn’t say entropy is a property of a system, but a property of an observer who describes a system.”,,,
https://www.quantamagazine.org/quantum-thermodynamics-revolution/

“Entropy is a property of an observer who describes a system.”,,????

Holy schizophrenic map Batman!!! Tell me it ain’t so!

Seems, as far as materialistic presuppositions are concerned, the thermodynamic ‘territory’ is far more concerned with the existence of the “map” of immaterial information, and even with the existence of the immaterial mind, than it ought to be concerned.

Aside from thermodynamics, in quantum mechanics the situation gets worse for atheistic materialists. Much worse!

As (the atheist) Steven Weinberg states, (in quantum mechanics) humans are brought into the laws of nature at the most fundamental level.,,, the instrumentalist approach (in quantum mechanics) turns its back on a vision that became possible after Darwin, of a world governed by impersonal physical laws that control human behavior along with everything else.,,, In quantum mechanics these probabilities do not exist until people choose what to measure,,, Unlike the case of classical physics, a choice must be made,,,

The Trouble with Quantum Mechanics – Steven Weinberg – January 19, 2017
Excerpt: The instrumentalist approach,, (the) wave function,, is merely an instrument that provides predictions of the probabilities of various outcomes when measurements are made.,,
In the instrumentalist approach,,, humans are brought into the laws of nature at the most fundamental level. According to Eugene Wigner, a pioneer of quantum mechanics, “it was not possible to formulate the laws of quantum mechanics in a fully consistent way without reference to the consciousness.”11
Thus the instrumentalist approach turns its back on a vision that became possible after Darwin, of a world governed by impersonal physical laws that control human behavior along with everything else. It is not that we object to thinking about humans. Rather, we want to understand the relation of humans to nature, not just assuming the character of this relation by incorporating it in what we suppose are nature’s fundamental laws, but rather by deduction from laws that make no explicit reference to humans. We may in the end have to give up this goal,,,
Some physicists who adopt an instrumentalist approach argue that the probabilities we infer from the wave function are objective probabilities, independent of whether humans are making a measurement. I don’t find this tenable. In quantum mechanics these probabilities do not exist until people choose what to measure, such as the spin in one or another direction. Unlike the case of classical physics, a choice must be made,,,
http://www.nybooks.com/article.....mechanics/

Again, (given materialistic presuppositions), the ‘territory’ is far more concerned with how I might read the map than it ought to be.

Here is a supplemental quote from Berlinski that may be of interest to some:

An Interview with David Berlinski – Jonathan Witt
Berlinski: There is no argument against religion that is not also an argument against mathematics. Mathematicians are capable of grasping a world of objects that lies beyond space and time….
Interviewer:… Come again(?) …
Berlinski: No need to come again: I got to where I was going the first time. The number four, after all, did not come into existence at a particular time, and it is not going to go out of existence at another time. It is neither here nor there. Nonetheless we are in some sense able to grasp the number by a faculty of our minds. Mathematical intuition is utterly mysterious. So for that matter is the fact that mathematical objects such as a Lie Group or a differentiable manifold have the power to interact with elementary particles or accelerating forces. But these are precisely the claims that theologians have always made as well – that human beings are capable by an exercise of their devotional abilities to come to some understanding of the deity; and the deity, although beyond space and time, is capable of interacting with material objects.
http://tofspot.blogspot.com/20.....-here.html

Of course, Godel incompleteness theorem can also be thrown into the mix, but for me, i.e. the incompleteness of math, that would just further accentuate Berlinski’s claim

75. 75
ET says:

Ed George:

The rate a rock falls will depend on gravity…

Gravity exists because of mathematics, that is gravity is a mathematical construct put into motion (pun intended).

76. 76
Nonlin.org says:

kairosfocus @36+

My point was simply that you have no way of knowing “true in all possible worlds” and if so, “necessity” is an unsupported claim. Examples: we falsely thought for a long time that “swans are white in all possible worlds” and that “sum of triangle angles is 180 deg in all possible worlds”.

If you want to say “all unmarried males are unmarried” do so. But that’s a tad silly and different from “all bachelors are unmarried”.

77. 77
kairosfocus says:

NL, The issue is the power of logic of being and particularly of that tied to structure and quantity. Perhaps the easiest case is something impossible of being, a square circle. The core characteristics to be squarish and circular stand in such contradiction that there is no possible world where such could exist. By contrast, if something is part of the framework for a — any — world, it will obtain in every world. For example on distinct identity A vs ~A, twoness must exist in any world as was discussed above. That is no distinct world is possible without distinct identity thus A vs not A and so also twoness. Because of this approach, we do not need to observe every world or observe it. As always, the issue pivots on first principles of right reason. KF

PS: Equivocation that distorts obvious meaning is fallacious.

78. 78
EricMH says:

@KF, it does not seem we are disagreeing. Constraint is not the same as causality. The laws of math entail certain things can and cannot happen, but do not entail what does happen.

79. 79
kairosfocus says:

EMH, mathematical properties clearly are not active, enabling causes. They often specify necessary conditions or constraints. Of course, this points to our modern understanding of causes as opposed to say Aristotle’s which is more like explanation. KF

80. 80
kairosfocus says:

EG, I just remind, Einstein traced gravitation to the warping of the spacetime fabric . . . the geometry of space . . . by massive bodies. That is how even light is bent through gravitational lensing. The warping of the geometry of spacetime is about as mathematically embedded a concept as we can get. KF

81. 81
Ed George says:

KF, you are using examples of physical interactions that we have been able to model mathematically as evidence that mathematics is inherent in the universe. Seems like a circular argument to me.

82. 82
kairosfocus says:

EG, do you not see that the very concepts you refer to, such as falling, rates, even gravitation are deeply, inextricably pervaded with issues of structure and quantity such that the neat separation you imagine between physical reality and mathematical models is impossible? That is what I have repeatedly shown, and it is what you have repeatedly ignored. That inattention is by now diagnostic. Also, it should by now be clear that reality is deeply embedded with structural and quantitative aspects so that the substance of reality is intensely mathematical. We do form mathematical models of systems, which are provisional, but that is only part of the story. This leads me to ponder the cultural forces that lead to such an intense struggle to dichotomise the physical and the mathematical, noting that in reality up to C19 Physicists and Mathematicians were on the whole pretty much the same people freely going back and forth between the empirical and the analytical. So far, that keeps on coming up, Kant. KF

83. 83
Ed George says:

KF

EG, do you not see that the very concepts you refer to, such as falling, rates, even gravitation are deeply, inextricably pervaded with issues of structure and quantity such that the neat separation you imagine between physical reality and mathematical models is impossible?

The separation is impossible because the mathematical models we have developed accurately model the physical reality. The horse always precedes the cart.

That is what I have repeatedly shown, and it is what you have repeatedly ignored. That inattention is by now diagnostic.

Not found compelling is not the same as ignored. If you are going to be so dismissive and insulting, I don’t think this discussion is worth my time and effort. Please show me the respect I have shown you.

Also, it should by now be clear that reality is deeply embedded with structural and quantitative aspects so that the substance of reality is intensely mathematical.

I agree. There is order in the universe and we are good at modelling it mathematically. But that doesn’t mean that mathematics exists without humans. A crystal can be modelled to the finest detail mathematically. But the diamond forms independent of the math and dependent on atomic structure and how it reacts to temperature and pressure.

84. 84
kairosfocus says:

EG, the phenomena in question are inextricably part of the matrix of reality: space, time, change, rates of action, and more. These and many others are inherently structural and quantitative. This is why the substance of that rational framework of structure and quantity is integral to reality. This is why it is appropriate for us to respond using our rational responsible freedom, studying the logic of structure and quantity and how it is embedded in reality as key facets of our culture. Err though we may often do in that process. And so, again and again I hear the echo of the Kantian ugly gulch when I find you trying to sever the inextricably entangled. So, I again point you to F H Bradley’s corrective — which you have never acknowledged. Namely, s/he who imagines that things in themselves are unknowable, across the gulch, by that postulate imply a claimed knowledge of such extramental reality. Thus, self-referential incoherence. We may err but we also can credibly know reality as our rational faculties reach out to the world we inhabit. And indeed a first self-evident truth is that error exists. So, truth exists and in some cases is knowable to undeniable certainty. In your latest case, the metastable diamond structure emerges through pressure and temperature interacting with the structural and quantitative, quantum properties of C-atoms. Where, pressure is a metric of force per unit area and temperature one of the average random energy per degree of microscopic freedom for a body or system. These concepts are deeply pervaded with issues of structure and quantity, which are thus mathematical; diamonds do not, cannot form apart from these inherently mathematical factors; something which is normally studied as part of any undergraduate programme in physics. Again, the grips of a cultural process are patently at work; one that seems to be coming in the main from trends in the arts in recent decades, as its influences are evidently unaware of relevant physics and its inextricably mathematical aspects. The physics is not on trial, nor is the mathematical framework of reality. Responsiveness to such is, and finding the relevant facts unpersuasive is sadly diagnostic. KF

PPS: As a reminder, Bradley, 1897:

We may agree, perhaps, to understand by metaphysics an attempt to know reality as against mere appearance, or the study of first principles or ultimate truths, or again the effort to comprehend the universe, not simply piecemeal or by fragments, but somehow as a whole [–> i.e. the focus of Metaphysics is critical studies of worldviews] . . . .

The man who is ready to prove that metaphysical knowledge is wholly impossible . . . himself has, perhaps unknowingly, entered the arena . . . To say the reality is such that our knowledge cannot reach it, is a claim to know reality ; to urge that our knowledge is of a kind which must fail to transcend appearance, itself implies that transcendence. For, if we had no idea of a beyond, we should assuredly not know how to talk about failure or success. And the test, by which we distinguish them, must obviously be some acquaintance with the nature of the goal. Nay, the would-be sceptic, who presses on us the contradictions of our thoughts, himself asserts dogmatically. For these contradictions might be ultimate and absolute truth, if the nature of the reality were not known to be otherwise . . . [such] objections . . . are themselves, however unwillingly, metaphysical views, and . . . a little acquaintance with the subject commonly serves to dispel [them]. [Appearance and Reality, 2nd Edn, 1897 (1916 printing), pp. 1 – 2; INTRODUCTION. At Web Archive.]

85. 85
Ed George says:

KF

The physics is not on trial, nor is the mathematical framework of reality. Responsiveness to such is, and finding the relevant facts unpersuasive is sadly diagnostic. KF

The fact that you can’t tolerate respectful disagreement without resorting to personal attack, you will forgive me for desiring not to pursue any future discussion with you. Life is too short. I wish you well.

86. 86
kairosfocus says:

EG, by twisting a question of diagnosis of trends in the history of ideas — specifically, the evident influence of Kant’s ugly gulch — into a claimed ad hominem you just went beyond the pale of civil discourse. Kindly, walk back. It remains, that the concepts, patterns and observations you appealed to to try to drive a wedge between physics and mathematics have consistently turned out to be pervaded by issues of structure and quantity. It is further clear from the above that you have been unresponsive to those clear facts. That calls for diagnosis and candidate number one is the influence of Kant’s error of imposing an unbridgeable gulch between our inner world (where we carry out our reflections on the logic of structure and quantity) and the outer one of things in themselves, where we readily see the pervasiveness of structure and quantity. The correlation between our rational reflections and the world we inhabit, experience, explore and observe does point to that world having a rational structure tracing to the same source as our minds. That’s an onward issue, the question that lags from posts 1, 2 and 3 in the series is that we need to ponder the mathematical aspect of reality. That aspect starts with yet another issue that requires appropriate response: for there to be a distinct world, there must be distinct identity, thus we see W = {A|~A} thence duality, unity, nullity and also the emergence of the natural numbers and things tied thereto. As JAD pointed out, the naturals include primes, which are a classic example of DISCOVERY rather than invention by human creativity. Similarly, Pythagoras’ generalisation of the 3-4-5 result and the like led to the shocking discovery of irrationals. Mathematicians explore logic model worlds indeed but in so doing they often encounter necessarily existent entities that then must appear in all possible worlds, precisely because they are part of the framework for any world to exist. KF

87. 87
kairosfocus says:

F/N: Let us take a key observation:

There is order in the universe and we are good at modelling it mathematically. But that doesn’t mean that mathematics exists without humans.

Here we see the concept that mathematics is essentially a human practice, a study. The contrast, there is order in the cosmos then uses a shift in terminology that misses a key aspect of that order — it is in material part inherently quantitative and structural. That is, it is of the substance that we may appropriately call mathematical.

Not that the labelling creates the reality as an inner phenomenon locked away from the world of things in themselves by the ugly gulch, that collective labelling (see the slippery slope to solipsism?) is a response to discovery.

We must face that, starting with that once any distinct world is, that means that there is inherently a contrast between some A and what is not A, ~A. Existence as a distinct physical entity or even as an idea immediately has as corollary, distinction from what is not the same. We therefore see existence vs non-existence, nullity. We see existence, unity. We see contrast thus duality. The numbers 0, 1, 2 swim into view, not the labels [numerals] we happen to find convenient, the substance.

This surfaces the question, what is truth. Again, not the label but the substance. To which Aristotle’s response speaks: truth says of what is that it is; and of what is not, that it is not. Truth accurately describes reality. It is attainable by us in key cases, e.g. error exists is undeniably true. And as that is so, knowledge exists as warranted, reliable truth that we acknowledge [to know, one has to accept, to actually believe], in this case to utter certainty.

Of course, nowadays, it is a struggle to form a coherent concept of truth, warrant, knowledge. That is how impoverished our sadly suicidally bent civilisation has become. And of course, yes these are going to seem strange and will be hard or even bitter to swallow. That is going to take time.

Now, back to the mathematical substance of reality. We see from distinct identity of a world, that the natural numbers are necessarily substantially present given the force of that identity. But, isn’t all of this just in our heads, if we aren’t here to do math it vanishes for want of sufficiently complex brains, poof.

Nope.

|| + ||| –> |||||

obtains whether we are there to perceive and contemplate it or not. Two is even and the first prime. Three is prime simply as it cannot be evenly shared in a whole number of slots apart from by ones. And so forth, property after property. We come along, discover significance, label, embed in systems of thought. But all of that is in response to substance that we find it necessary to accurately describe.

We develop symbols:

2 + 3 = 5

Those are cultural, convenient, helpful. But they are not the substance, they are our way to handle that substance. A substance of reality that is quantitative (amenable to measure) and structural (bound up in coherent relationships). We label such, Mathematics. That does not create a discipline by the poof-magic of words, it is a way for us to refer to the substantial reality and to our explorations and development.

Nope, views must be accountable to substantial reality. That substance starts with there being a distinct world. We are seeing something that is objectively true, credibly accurately describing realities that are beyond being figments of imagination such as Mr Spock or Fr Brown or Tolkein’s world and rings.

Beyond numbers we may see the continuum, the rock falls through space. But what is space. We see here a quantitative structure tied to length and dimensionality. There is more or less of length and it can be in different directions. To get there conceptually, we need integers (thus negative numbers), we need fractions, we need power series that can sum to infinite numbers of terms and which converge in some cases, as we discovered — nope, it was not an invention — the irrationals. Sqrt-2 deserves to be a famous number once it was seen that the diagonal of a square was inherently in-commensurable with its sides. Imagine, this was actually seen as of such significance that it was viewed with religious awe.

Yes, they saw more truly than we now do in too many cases, this sees into the roots of reality.

Likewise, 0 = 1 + e^i*pi should not only give high confidence in the coherence of vast domains in maths, but it should give us a view into the roots of reality. Another day.

Just to get to understand the space in which a stone falls, we have seen a huge body of quantitative structures embedded inextricably in reality. The substance is there. Bring on the time for a trajectory and we see the same continuum again.

Then, to get to the substance of the gravitation that is at root of the force that triggers the accelerated motion, we find not only further structures and quantities such as displacement x, velocity dx/dt and acceleration dv/dt as well as the panoply of infinitesimals, limits etc, but also the warping of spacetime through the influence of a massive body, a planet. (A ship or a mountain shows similar effects, on a smaller scale.)

Then, looking through telescopes, we find that gravitation is central to the grand structure and function of the overall world we inhabit.

Reality is pervaded with a rationally accessible framework that is quantitative and structural.

Nor does the struggle we may have to acknowledge that pervasive substance change it. It simply forces us to face the need to pursue comparative difficulties and recognise what is a superior explanation. It is simply not good enough to say, oh yes order without acknowledging the specific type: structural, quantitative — that is, mathematical.

And so, we come to Newton, in his General Scholium to Principia; who was moved to an exercise in philosophical theology as he pondered his discoveries:

. . . This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centres of other like systems, these, being formed by the like wise counsel, must be all subject to the dominion of One; especially since the light of the fixed stars is of the same nature with the light of the sun, and from every system light passes into all the other systems: and lest the systems of the fixed stars should, by their gravity, fall on each other mutually, he hath placed those systems at immense distances one from another.

This Being governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont to be called Lord God pantokrator , or Universal Ruler; for God is a relative word, and has a respect to servants; and Deity is the dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without dominion, cannot be said to be Lord God; for we say, my God, your God, the God of Israel, the God of Gods, and Lord of Lords; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods; we do not say, my Infinite, or my Perfect: these are titles which have no respect to servants. The word God usually signifies Lord; but every lord is not a God. It is the dominion of a spiritual being which constitutes a God: a true, supreme, or imaginary dominion makes a true, supreme, or imaginary God. And from his true dominion it follows that the true God is a living, intelligent, and powerful Being; and, from his other perfections, that he is supreme, or most perfect. He is eternal and infinite, omnipotent and omniscient; that is, his duration reaches from eternity to eternity; his presence from infinity to infinity; he governs all things, and knows all things that are or can be done. He is not eternity or infinity, but eternal and infinite; he is not duration or space, but he endures and is present. He endures for ever, and is every where present; and by existing always and every where, he constitutes duration and space. Since every particle of space is always, and every indivisible moment of duration is every where, certainly the Maker and Lord of all things cannot be never and no where. Every soul that has perception is, though in different times and in different organs of sense and motion, still the same indivisible person. There are given successive parts in duration, co-existent puts in space, but neither the one nor the other in the person of a man, or his thinking principle; and much less can they be found in the thinking substance of God. Every man, so far as he is a thing that has perception, is one and the same man during his whole life, in all and each of his organs of sense. God is the same God, always and every where. He is omnipresent not virtually only, but also substantially; for virtue cannot subsist without substance. In him are all things contained and moved [i.e. cites Ac 17, where Paul evidently cites Cleanthes]; yet neither affects the other: God suffers nothing from the motion of bodies; bodies find no resistance from the omnipresence of God. It is allowed by all that the Supreme God exists necessarily; and by the same necessity he exists always, and every where. [i.e accepts the cosmological argument to God.] Whence also he is all similar, all eye, all ear, all brain, all arm, all power to perceive, to understand, and to act; but in a manner not at all human, in a manner not at all corporeal, in a manner utterly unknown to us. As a blind man has no idea of colours, so have we no idea of the manner by which the all-wise God perceives and understands all things. He is utterly void of all body and bodily figure, and can therefore neither be seen, nor heard, or touched; nor ought he to be worshipped under the representation of any corporeal thing. [Cites Exod 20.] We have ideas of his attributes, but what the real substance of any thing is we know not. In bodies, we see only their figures and colours, we hear only the sounds, we touch only their outward surfaces, we smell only the smells, and taste the savours; but their inward substances are not to be known either by our senses, or by any reflex act of our minds: much less, then, have we any idea of the substance of God. We know him only by his most wise and excellent contrivances of things, and final cause [i.e from his designs]: we admire him for his perfections; but we reverence and adore him on account of his dominion: for we adore him as his servants; and a god without dominion, providence, and final causes, is nothing else but Fate and Nature. Blind metaphysical necessity, which is certainly the same always and every where, could produce no variety of things. [i.e necessity does not produce contingency] All that diversity of natural things which we find suited to different times and places could arise from nothing but the ideas and will of a Being necessarily existing. [That is, implicitly rejects chance, Plato’s third alternative and explicitly infers to the Designer of the Cosmos.] But, by way of allegory, God is said to see, to speak, to laugh, to love, to hate, to desire, to give, to receive, to rejoice, to be angry, to fight, to frame, to work, to build; for all our notions of God are taken from. the ways of mankind by a certain similitude, which, though not perfect, has some likeness, however. And thus much concerning God; to discourse of whom from the appearances of things, does certainly belong to Natural Philosophy.

Yes, the mathematical substance that we find inextricably intertwined in the fabric of reality may be disturbing and may challenge us to ponder its worldviews significance. Perhaps, that was intended by the One who framed it.

But, that is an onward question, the matter before us is the need to acknowledge that rationally accessible framework of structure and quantity.

KF

88. 88
bornagain77 says:

Ed George holds,,,

There is order in the universe and we are good at modelling it mathematically. But that doesn’t mean that mathematics exists without humans.

Ed George is basically saying that mathematics is invented by humans instead of being discovered by humans.

Yet as was mentioned previously,

“However, if Pythagoras had not lived, someone else would have discovered exactly the same Pythagoras theorem. Moreover, that theorem means the same to us today as it meant to Pythagoras 2,500 years ago”.
– Edward Frenkel

And indeed there are no perfect triangles within the universe, yet we know for a fact that triangles exist independently of human experience.

Naturalism and Self-Refutation – Michael Egnor – January 31, 2018
Excerpt: Mathematics is certainly something we do. Is mathematics “included in the space-time continuum [with] basic elements … described by physics”? It seems a stretch. What is the physics behind the Pythagorean theorem? After all, no actual triangle is perfect, and thus no actual triangle in nature has sides such that the Pythagorean theorem holds. There is no real triangle in which the sum of the squares of the sides exactly equals the square of the hypotenuse. That holds true for all of geometry. Geometry is about concepts, not about anything in the natural world or about anything that can be described by physics. What is the “physics” of the fact that the area of a circle is pi multiplied by the square of the radius? And of course what is natural and physical about imaginary numbers, infinite series, irrational numbers, and the mathematics of more than three spatial dimensions? Mathematics is entirely about concepts, which have no precise instantiation in nature,,
https://evolutionnews.org/2018/01/naturalism-and-self-refutation/

Moreover, as Dr. Egnor’s article alluded to, Mathematics itself exists in a transcendent, beyond space and time, realm which is not reducible any possible material explanation. This transcendent mathematical realm has been referred to as a Platonic mathematical world.

Platonic mathematical world – image
https://image.slidesharecdn.com/quantuminformation2-120301000431-phpapp01/95/quantum-information-14-728.jpg?cb=1330561190

In fact, ‘Mathematical Platonism enjoys widespread support and is frequently considered the default metaphysical position with respect to mathematics.,,,

Mathematical Platonism
Excerpt: Mathematical platonism enjoys widespread support and is frequently considered the default metaphysical position with respect to mathematics. This is unsurprising given its extremely natural interpretation of mathematical practice. In particular, mathematical platonism takes at face-value such well known truths as that “there exist” an infinite number of prime numbers, and it provides straightforward explanations of mathematical objectivity and of the differences between mathematical and spatio-temporal entities. Thus arguments for mathematical platonism typically assert that in order for mathematical theories to be true their logical structure must refer to some mathematical entities, that many mathematical theories are indeed objectively true, and that mathematical entities are not constituents of the spatio-temporal realm.
The most common challenge to mathematical platonism argues that mathematical platonism requires an impenetrable metaphysical gap between mathematical entities and human beings. Yet an impenetrable metaphysical gap would make our ability to refer to, have knowledge of, or have justified beliefs concerning mathematical entities completely mysterious. Frege, Quine, and “full-blooded platonism” offer the three most promising responses to this challenge.,,,
http://www.iep.utm.edu/mathplat/

The denial of the independent “platonic’ existence of mathematics is an interesting position for Ed George to hold.

Ed George has previously claimed, on UD, to be a Christian, and yet the only people who have a philosophical predisposition to deny the independent existence of mathematics are reductive materialists, i.e. Darwinists.

In fact, Darwinists deny the independent reality of any immaterial, abstract, concepts. (such as soul, mind, personhood, justice, mercy, etc.. etc..)

For instance, the concept of species itself is an abstract, immaterial, concept that can find no grounding within atheistic materialism:,,,
, As was touched upon,, the primary failing of atheistic materialism is that it denies the existence of the human soul altogether.
In other words, the entire abstract, immaterial, concept of “you” existing as a real person can find no grounding within atheistic materialism.
There simply is no “me”, “myself”, or “I” within the reductive materialism of the Darwinian worldview for atheists to appeal to:
https://uncommondescent.com/intelligent-design/how-does-human-language-differ-from-animal-signals/#comment-669624

What is interesting is that although Darwinists, (and Ed George), deny that this Platonic mathematical realm exists, Darwinists need this transcendent world of mathematics to exist in order for their theory to even be considered scientific in the first place.

Mathematics literally provides the backbone for all of science, engineering and technology

A VIEW OF MATHEMATICS – Alain Connes
Excerpt: Mathematics is the backbone of modern science,,,
http://www.alainconnes.org/docs/maths.pdf

The predicament that Darwinists, (and Ed George), find themselves in regards to denying the reality of this transcendent, immaterial, world of mathematics, and yet needing validation from this transcendent, immaterial, world of mathematics in order to be considered scientific in the first place, should be the very definition of a self-refuting worldview,,, (then again, the denial of free will by Darrwinists would rank right alongside the denial of mathematics as a self-refuting worldview).

What Does It Mean to Say That Science & Religion Conflict? – M. Anthony Mills – April 16, 2018
Excerpt: In fact, more problematic for the materialist than the non-existence of persons is the existence of mathematics. Why? Although a committed materialist might be perfectly willing to accept that you do not really exist, he will have a harder time accepting that numbers do not exist. The trouble is that numbers — along with other mathematical entities such as classes, sets, and functions — are indispensable for modern science. And yet — here’s the rub — these “abstract objects” are not material. Thus, one cannot take science as the only sure guide to reality and at the same time discount disbelief in all immaterial realities.
https://www.realclearreligion.org/articles/2018/04/16/what_does_it_mean_to_say_that_science_and_religion_conflict.html

Moreover, the default assumption behind the quest to find a quote unquote “Theory of Everything” is that there really is some type of unifying (Platonic) mathematical structure behind the universe to be discovered, not invented, by man.

In fact, both Einstein, who ‘discovered’ general relativity, and Eugene Wigner, who’s insights into Quantum Mechanics continue to bear fruit for quantum mechanics (Zeilinger), are both on record as regarding it as a epistemological miracle that we are able to accurately model the universe with mathematics,,,

On the Rational Order of the World: a Letter to Maurice Solovine – Albert Einstein – March 30, 1952
Excerpt: “You find it strange that I consider the comprehensibility of the world (to the extent that we are authorized to speak of such a comprehensibility) as a miracle or as an eternal mystery. Well, a priori, one should expect a chaotic world, which cannot be grasped by the mind in any way .. the kind of order created by Newton’s theory of gravitation, for example, is wholly different. Even if a man proposes the axioms of the theory, the success of such a project presupposes a high degree of ordering of the objective world, and this could not be expected a priori. That is the ‘miracle’ which is constantly reinforced as our knowledge expands.
There lies the weakness of positivists and professional atheists who are elated because they feel that they have not only successfully rid the world of gods but “bared the miracles.”
-Albert Einstein
http://inters.org/Einstein-Letter-Solovine

The Unreasonable Effectiveness of Mathematics in the Natural Sciences – Eugene Wigner – 1960
Excerpt: ,,certainly it is hard to believe that our reasoning power was brought, by Darwin’s process of natural selection, to the perfection which it seems to possess.,,,
It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them.,,,
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
http://www.dartmouth.edu/~matc.....igner.html

89. 89
bornagain77 says:

And indeed, it is a miracle. Our ability to even apply math to the world in the first place is dependent on the universe being exceptionally, even miraculously, ‘flat’.

The reason why Euclidean (3 Dimensional) geometry is even applicable in our science, technology, and engineering in the first place is because the 4-Dimensional space-time of our universe (General Relativity) is exceptionally, and unexpectedly “flat”. As Fraser Cain stated in the following article, “We say that the universe is flat, and this means that parallel lines will always remain parallel. 90-degree turns behave as true 90-degree turns, and everything makes sense.,,, In fact, astronomers estimate that the universe must have been flat to 1 part within 1×10^57 parts.
Which seems like an insane coincidence.’

How do we know the universe is flat? Discovering the topology of the universe – by Fraser Cain – June 7, 2017
Excerpt: We say that the universe is flat, and this means that parallel lines will always remain parallel. 90-degree turns behave as true 90-degree turns, and everything makes sense.,,,
Since the universe is flat now, it must have been flat in the past, when the universe was an incredibly dense singularity. And for it to maintain this level of flatness over 13.8 billion years of expansion, in kind of amazing.
In fact, astronomers estimate that the universe must have been flat to 1 part within 1×10^57 parts.
Which seems like an insane coincidence.
https://phys.org/news/2017-06-universe-flat-topology.html

Why We Need Cosmic Inflation
By Paul Sutter, Astrophysicist | October 22, 2018
Excerpt: As best as we can measure, the geometry of our universe appears to be perfectly, totally, ever-so-boringly flat. On large, cosmic scales, parallel lines stay parallel forever, interior angles of triangles add up to 180 degrees, and so on. All the rules of Euclidean geometry that you learned in high school apply.
But there’s no reason for our universe to be flat. At large scales it could’ve had any old curvature it wanted. Our cosmos could’ve been shaped like a giant, multidimensional beach ball, or a horse-riding saddle. But, no, it picked flat.
https://www.space.com/42202-why-we-need-cosmic-inflation.html

Simply put, without some remarkable degree of exceptional, and stable, flatness for the universe, (as well as exceptional stability for all the other constants), Euclidean (3-Dimensional) geometry would not be applicable to our world. or to the universe at large, and this would make science and engineering for humans, for all practical purposes, impossible.

Scientists Question Nature’s Fundamental Laws – Michael Schirber – 2006
Excerpt: “There is absolutely no reason these constants should be constant,” says astronomer Michael Murphy of the University of Cambridge. “These are famous numbers in physics, but we have no real reason for why they are what they are.”
The observed differences are small-roughly a few parts in a million-but the implications are huge (if they hold up): The laws of physics would have to be rewritten, not to mention we might need to make room for six more spatial dimensions than the three that we are used to.”,,,
The speed of light, for instance, might be measured one day with a ruler and a clock. If the next day the same measurement gave a different answer, no one could tell if the speed of light changed, the ruler length changed, or the clock ticking changed.
http://www.space.com/2613-scie.....-laws.html

In other words, it literally is a “1 in 10^57 miracle” that we are even able to apply Euclidean mathematics to the world in the first place in order to build technologically advanced civilizations.

Job 38:4-5
“Where were you when I laid the earth’s foundation?
Tell me, if you understand.
Who marked off its dimensions? Surely you know!
Who stretched a measuring line across it?

And indeed infusing math and logic, in a top down fashion, onto material substrates, has enabled the ‘miracle’ of modern technology,,,, (to repeat part of post 74)

Describing Nature With Math By Peter Tyson – Nov. 2011
Excerpt: Indeed, many of the technologies you and I enjoy every day simply would not work without mathematics. When you do a Google search, you’re relying on 19th-century algebra, on which the search engine’s algorithms are based. When you watch a movie, you may well be seeing mountains and other natural features that, while appearing as real as rock, arise entirely from mathematical models. When you play your iPod, you’re hearing a mathematical recreation of music that is stored digitally; your cell phone does the same in real time.
“When you listen to a mobile phone, you’re not actually hearing the voice of the person speaking,” Devlin told me. “You’re hearing a mathematical recreation of that voice. That voice is reduced to mathematics.”
http://www.pbs.org/wgbh/nova/p.....-math.html

Nor, if the universe were not exceptionally flat, would we have ever been able to eventually discover the higher dimensional mathematics that lay behind our most accurate theories in science.

Quantum Mechanics, Special Relativity, General Relativity and Christianity – video

In summary, Ed George’s denial of the independent reality of the immaterial world of mathematics, the “Platonic” world, is insane.

Of note: Ed George could have argued, via Godel, that the immaterial world of mathematics needs God to ‘breathe fire into the equations’, but that is not what he is arguing. He is instead arguing the absurd position that humans invent mathematics instead of discovering the hidden truths of a preexistent, immaterial, mathematical “Platonic”, world.

BRUCE GORDON: Hawking’s irrational arguments – October 2010
Excerpt: ,,,The physical universe is causally incomplete and therefore neither self-originating nor self-sustaining. The world of space, time, matter and energy is dependent on a reality that transcends space, time, matter and energy.
This transcendent reality cannot merely be a Platonic realm of mathematical descriptions, for such things are causally inert abstract entities that do not affect the material world,,,
Rather, the transcendent reality on which our universe depends must be something that can exhibit agency – a mind that can choose among the infinite variety of mathematical descriptions and bring into existence a reality that corresponds to a consistent subset of them. This is what “breathes fire into the equations and makes a universe for them to describe.” Anything else invokes random miracles as an explanatory principle and spells the end of scientific rationality.,,,
Universes do not “spontaneously create” on the basis of abstract mathematical descriptions, nor does the fantasy of a limitless multiverse trump the explanatory power of transcendent intelligent design. What Mr. Hawking’s contrary assertions show is that mathematical savants can sometimes be metaphysical simpletons. Caveat emptor.
http://www.washingtontimes.com.....arguments/

90. 90
kairosfocus says:

BA77, we must reckon with the effects of our education and cultural immersion. That can make it very hard to see that the admitted order in the world is substantially — not just culturally — quantitative and structural, being amenable to rational inquiry. Even, in the face of massive facts on the matter. That is how far gone our civilisation is. Sad. KF

91. 91
ET says:

How does Ed George think the universe was intelligently designed without the use of mathematics?

92. 92
Ed George says:

ET

How does Ed George think the universe was intelligently designed without the use of mathematics?

I can make a snowman without using mathematics. Surely God doesn’t require something as mundane as mathematics to create.

93. 93
ET says:

Ed George:

I can make a snowman without using mathematics.

Cuz a snowman is so much like the universe. Clearly you don’t have a clue. You can make a snowman because mathematics permeates the universe.

Surely God doesn’t require something as mundane as mathematics to create.

Cuz you noes. Pathetic. In the words of YECs God Created the universe using relativity, ie a mathematical concept

94. 94
kairosfocus says:

EG, that snowman works because of a host of quantitative and structural properties starting with temperature and its effects on water molecules, plus the form factor of the H2O molecule. Those are substantially mathematical, whether or not you are inclined to agree that such is the case. KF

95. 95
Nonlin.org says:

kairosfocus,

I support logic as our best tool, but will not go as far as saying “in all possible worlds”.

You say: “the easiest case is something impossible of being, a square circle”. What about triangles in curved space (sum of angles = 180 deg)? Same story?

Regardless, my strongest objection is to ‘necessity’ as alternative to ‘design’ in nature, not to married bachelors. Why don’t you address that? My assertion is that the ONLY ‘necessity’ instances we can confirm are rules imposed by design.

96. 96
Nonlin.org says:

kairosfocus,

To dispute “all possible worlds”, let’s take another example: 2+2=4. You mean to tell me that if I put 2 male and 2 female dogs in a black box, I will ALWAYS have 4 dogs when opening? What about 2 apples + 2 pears? Will that make 4 apples? 4 pears? This is not equivocation, it’s real life that you assume can be put in a straitjacket. But it can’t. Modern physics is full of such examples.

97. 97
kairosfocus says:

NL: Equivocating the meaning of square and circular does not answer the question. Next, mechanical necessity does not account for high contingency, only chance and/or intelligence can; cf. Newton on this. After that, equivocating the sense of the + operator only shows the underlying problem. || + ||| –> |||||, symbolised 2 + 3 = 5 speaks for itself. Dogs interacting reproductively is a radically different process. Adding goes to the common denominator, 2 apples + 2 pears gives 4 fruit or even objects. Modern Physics relies on the distinct identity of mathematical symbols and operations to derive its results. KF

98. 98
Nonlin.org says:

kairosfocus,

Again, you wrongfully substitute theory to reality (which you improperly call “equivocation”). If you haven’t noticed, like “evolution”, Modern Physics is stuck. And that’s due precisely to the differences between theory and reality.

What does this mean: “mechanical necessity does not account for high contingency, only chance and/or intelligence can”?

99. 99
kairosfocus says:

NL, language is inherently ambiguous so it is necessary to consider context. In this case, circles and squares are planar figures in the context of a familiar space commonly termed Euclidean. When one switches context without due attention or notice, an equivocation creates an apparent impossibility of incoherence but it is due to context switching. And that is not a substitution of theory for reality. Next, mechanical necessity speaks to cases where once initial conditions and dynamics are set, a train of consequent stages unfolds in a precise determined trajectory [in phase space]. When under closely similar initial conditions highly variable outcomes can and do happen, this is accounted for on chance (think, tumbling die) and/or intelligently directed configuration, e.g. we may set a die to a reading or may so load it that its outcome follows a preferred distribution. KF

100. 100
kairosfocus says:

F/N: IEP on Necessary vs Contingent etc, using possible words speak:

https://www.iep.utm.edu/apriori/#H3

>>3. The Necessary/Contingent Distinction

A necessary proposition is one the truth value of which remains constant across all possible worlds. Thus a necessarily true proposition is one that is true in every possible world, and a necessarily false proposition is one that is false in every possible world. By contrast, the truth value of contingent propositions is not fixed across all possible worlds: for any contingent proposition, there is at least one possible world in which it is true and at least one possible world in which it is false.

The necessary/contingent distinction is closely related to the a priori/a posteriori distinction. It is reasonable to expect, for instance, that if a given claim is necessary, it must be knowable only a priori. Sense experience can tell us only about the actual world and hence about what is the case; it can say nothing about what must or must not be the case. Contingent claims, on the other hand, would seem to be knowable only a posteriori, since it is unclear how pure thought or reason could tell us anything about the actual world as compared to other possible worlds.

While closely related, these distinctions are not equivalent. The necessary/contingent distinction is metaphysical: it concerns the modal status of propositions. As such, it is clearly distinct from the a priori/a posteriori distinction, which is epistemological. Therefore, even if the two distinctions were to coincide, they would not be identical.

But there are also reasons for thinking that they do not coincide. Some philosophers have argued that there are contingent a priori truths (Kripke 1972; Kitcher 1980b). An example of such a truth is the proposition that the standard meter bar in Paris is one meter long. This claim appears to be knowable a priori since the bar in question defines the length of a meter. And yet it also seems that there are possible worlds in which this claim would be false (e.g., worlds in which the meter bar is damaged or exposed to extreme heat). Comparable arguments have been offered in defense of the claim that there are necessary a posteriori truths. Take, for example, the proposition that water is H2O (ibid.). It is conceivable that this proposition is true across all possible worlds, that is, that in every possible world, water has the molecular structure H2O. But it also appears that this proposition could only be known by empirical means and hence that it is a posteriori. Philosophers disagree about what to make of cases of this sort, but if the above interpretation of them is correct, a proposition’s being a priori does not guarantee that it is necessary, nor does a proposition’s being a posteriori guarantee that it is contingent.

Finally, on the grounds already discussed, there is no obvious reason to deny that certain necessary and certain contingent claims might be unknowable in the relevant sense. If indeed such propositions exist, then the analytic does not coincide with the necessary, nor the synthetic with the contingent.>>

KF

101. 101
102. 102
Nonlin.org says:

kf,

I said: “my strongest objection is to ‘necessity’ as alternative to ‘design’ in nature”

You say: “mechanical necessity speaks to cases where once initial conditions and dynamics are set, a train of consequent stages unfolds in a precise determined trajectory [in phase space]”

This doesn’t address my concern because:
– What is “initial” in biology, cosmology or anywhere in nature?
– How would you evaluate “precise determined trajectory”? There are ALWAYS intervening factors from “initial” to “final”.
– How would you “precisely” evaluate anything? What does “precisely” even mean (other than yet another theoretical concept)?
– Some form of Design is ALWAYS present in the “initial” conditions or on the trajectory to the “final” so ‘necessity’ is never an alternative to ‘design’.
– “true in all possible worlds” still doesn’t make any sense in ANY possible world 🙂

103. 103
kairosfocus says:

NL, mechanical necessity is one of the defaults in the explanatory filter. It is ruled out for a relevant aspect of a phenomenon based on high contingency of outcomes on closely similar initial conditions. Such high contingency then can be accounted for on chance and/or intelligently directed configuration. In the case where we have functionally coherent and configuration-specific complex [500 – 1000 bits+ of descriptive info] organisation and/or information, chance — the second default — is not plausible. That is how we reliably infer to design on FSCO/I. However, in aggregate, the behaviour of a system will manifest contributions of necessity, chance influences (often, “noise” or “random error”) and possibly design. The core of Newtonian dynamics lays out relationships of mechanical necessity: F = 0 => a = 0, F = dP/dt, with m = const F = ma, Action = – reaction. By mechanical necessity, given initial conditions lead to precise, predictable outcomes; e.g. the launch of a rocket and Tsiolkovsky’s eqns. Similarly, the classic projectile in vacuo, and we can bring in air resistance and still have deterministic trajectories, though the factors get much more complex. This applies to an ideal and extends to reality once chance disturbances are negligible. Otherwise, we feed them in, using sets of differential or difference equations. Sometimes things are too complex and we go to statistical mechanics or the like — consider a large collection of boxes with marbles set out on a standard grid then simultaneously push hard on LHS pistons with the same size impulse. After a time the marbles in the boxes would follow many diverse tracks but will follow Maxwell-Boltzmann statistics, due to all sorts of butterfly effect factors . . . which inject uncontrollable randomness. Going beyond that immediate focus, the world we inhabit shows fine tuning of the physics, the mechanical necessity under discussion is for a going concern world. I trust this helps, much more can be elaborated, this is indicative and uses toy examples. KF

104. 104
Nonlin.org says:

kf,

105. 105
kairosfocus says:

NL,

Perhaps, a glance at differential equation/ transfer function dynamics will help you, i/l/o the per aspect approach. Accordingly, I will append a basic block diagram of a dynamic-stochastic system in the OP above.

In addition, the train of objections shows one reason why it is too often necessary to be more expansive than brevity would wish for in this sort of highly polarised, multiple aspect debate. Remarks are shaped by years of seeing that sort of repeated objection.

Next, as it is historically a chief example, let us start from Newton’s first law of motion [= momentum, in modern language], the law of inertia: “a body will continue in its state of rest or of uniform, straight line motion, unless it is affected by an external, unbalanced force.”

That is, the inertial property of mass m is such that force F = 0 => a = 0; i.e. neither a change of speed nor of direction. Momentum, P once initially present (even P = 0) will not change UNLESS there is an external and unbalanced force acting. In turn this means in part that a body at rest may reflect a pattern of forces in balance as regards each axis of motion and rotation, but it may well have deflections and distortions that reflect stress-strain relationships starting with Hooke’s law etc. This first law also implies that any curvature in movement implies accelerated motion. (P = m*v.)

This is the first aspect of momentum, the steady state due to inertia. Where, “aspect” is a crucial conceptual term:

3. a way in which a thing may be regarded; interpretation; view.
4. part; feature; phase, as of a subject or problem . . .
6. view commanded; exposure: a house with a southern aspect.
7. the side or surface facing a given direction: the dorsal aspect of a fish. [RHK Webster’s]

That first aspect identifies a baseline, from which further aspects affect initial and onward motion and changes, i.e. dynamics.

The second, Dynamic Law then informs us regarding moving bodies: “Force is the rate of change of momentum.” Force, separately, is a push or pull which may distort and/or accelerate a body changing direction and/or speed. F = dP/dt, so if m = const, F = m*a. (A rocket is a relevant case of m != 0, thence Tsiolkovsky’s rocket equations.)

The third law on action and reaction, in expanded form: “bodies interact in pairs, and when they do so, they exert upon each other forces which are equal in size, opposite in direction and along the same line of action.” That is, F_a + F_r = 0, vectorially. The effects of these forces will vary, depending on relative masses, freedom to move etc.

This then sets up how the baseline changes, and indeed, differential equations for systems focus on input forcing functions, internal feedback and external results, including influences of noise [or stochastic aspects]. So, too, unless one separates out the underlying mechanical necessity at work (here F = dP/dt), one will be ill equipped to understand initial, intervening and onward states of affairs, trends and changes, disruptions etc. One of the reasons why classical writers did not get far with dynamics was that they did not isolate key aspects so they thought motion had to be forced, and onward thought in terms of a natural location, hence earth as disreputable sump of the cosmos. For example, understanding air resistance requires separating out unaccelerated motion, falling without air resistance and falling with air resistance. Where, later, lift-drag dynamics (which are very speed-dependent) will be material.

Understanding mechanical necessity is a first principle of dynamics.

Next, stochastic effects can trigger all sorts of random effects, ranging from being of minor importance to being utterly dominant. There have been sharp exchanges over “chance” here at UD too. The best answer is that we have quantum effects that seem inherently chance based and other forms of effective chance. Sometimes, a circumstance may be such that there is hyper-sensitivity to uncontrollable micro-variations, leading to chaos, e.g. a falling, tumbling die is in principle mechanical necessity but due to eight corners and twelve edges, the butterfly effect leads to an effectively unpredictable, uncontrolled outcome. Likewise, the clash of separately determined trains can produce statistical randomness. Classically, names are not random and telephone companies assign numbers in a planned way. But the two are uncorrelated so that the line codes can be used as a poor man’s random number table. Similarly, pi is an exactly determined number, but the value is not coordinated with the system that assigns decimal digits, so extended tables of pi can be used as random number tables.

The signatures of these two causal factors are that for a suitable circumstance, closely similar initial circumstances will exhibit low contingency or high contingency. A heavy die reliably falls at a known rate of acceleration. Once it tumbles and settles, the value is effectively unpredictable.

Thirdly, of course, we have intelligently directed configuration. A string of 1,000 coins can be set to read out 143 ASCII code characters expressing a meaningful statement. The blind mechanical and chance forces of the observed cosmos will not suffice to adequately explain such.

Now, it is ever easy to multiply sharpish questions and pose on dismissiveness or reducing the untutored interlocutor to confusion. That was in part what Socrates did, leading to his demise at the hands of his alienated compatriots. And while hemlock is usually not resorted to nowadays, there is the “don’t feed da trolls” reaction beyond a certain point. So, it is better to go to a dialogue model of mutual exploration.

So far, I have taken further time to show the importance of isolating key aspects of a situation and — where it is possible — using toy examples to tease out key factors that usually act together. That is the heart of the experimental method of investigation. Newton’s triumph is a salutary lesson.

Let me take a moment with your string of questions, i/l/o the above, noting that it takes time to find time to pause to address strings of such questions, which sometimes need to be deconstructed due to how loaded they are with prior contexts more than directly answered:

>>– What is “initial” in biology, cosmology or anywhere in nature?>>

1: what is temporally prior, at whatever start point makes sense. Big bang, solar dust cloud, Darwin’s warm pond, first cell based life, branch point for multicellular, etc.

2: the question is too general to be simply answered otherwise.

>>– How would you evaluate “precise determined trajectory”? There are ALWAYS intervening factors from “initial” to “final”.>>

3: Your recognition of intervening factors precisely shows the significance of understanding start-points and the trajectories they set.

4: Such trajectories are best studied on toy examples until reliable dynamics are in hand, then we may feed in complicating factors, such is standard science, engineering and modelling praxis.

5: The point where an intervening force acts is a secondary start point and the same dynamics allow for understanding its own impact, then we move to the second, third etc intervening forces at their points of action, including stochastic effects.

6: When we deal with continuum, we use calculus approaches to move to infinitesimal increments.

>>– How would you “precisely” evaluate anything? What does “precisely” even mean (other than yet another theoretical concept)?>>

7: Refer to Science and Engineering praxis, where precision here includes both reliability and accuracy up to some calibration curve. The SI system and traceable measurement back to standards is relevant. Not everything can be spelled out to the convenience of a commenter on his demand .

8: Dismissiveness regarding theory is grossly ill-advised.

>>– Some form of Design is ALWAYS present in the “initial” conditions or on the trajectory to the “final” so ‘necessity’ is never an alternative to ‘design’.>>

9: You have here built in a presumption of design. Detection of whether design is a credible, empirically evident cause is a key question of design theory, for excellent reason.

10: Chance and necessity, on the record since Plato in The Laws Bk X c 360 BC, have been viewed as alternatives to design.

>>– “true in all possible worlds” still doesn’t make any sense in ANY possible world>>

11: Possible worlds has been already discussed, and what would exist in some vs all such worlds also. Likewise, what is not possible of being in any possible world.

12: That summary of the logic of being is not on trial. Our response to it is.

KF

106. 106
Nonlin.org says:

KF,

Are you kidding me? Can’t you answer a few simple questions with simple answers? …without making a novel out of it?

1. You’re the one making a case for “initial” and now you say “whatever start point makes sense”? You should know this doesn’t work – see chaos theory. Big bang has not been observed but inferred based on current observations and a ton of assumptions – it is not a good “start point” for anything. And of course “Darwin’s warm pond” is just a fictional start of a fictive story.

2-6. Seriously? “Initial” condition is meaningless if unknown intervening factors are significant. You got it exactly backwards: “MY recognition of intervening factors precisely shows the INSIGNIFICANCE of understanding start-points and the trajectories they set”. See tornadoes – who cares about the trajectory set by the “initial condition” when we seldom can forecast more than one day trajectory ahead and that only by assuming no major unknown intervening factors (wrong most of the time)?

7-8. To the contrary, science and Engineering praxis shows you that there’s always a limit to “precisely”. Combine that with chaos theory that affects anything complex enough like biology/cosmology and you have exactly ZERO precision, hence no determinism, hence no necessity. I am not dismissing the theory – just pointing out that you’re applying it beyond its acceptable scope.

9. Not a presumption, but a logical analysis: http://nonlin.org/intelligent-design/

10. Good luck to Plato, but he’s wrong (not the first or last time) – see my analysis. We are debating “necessity” here, so your appeal to authority is undermining rather than supporting your argument.

11. Discussed – yes, convincing – no. You can’t seriously define something and then refer to it as “true in all possible worlds” just because you defined it so. I already discussed several counterexamples but you chose to label them as “Equivocating“ which really doesn’t fit: e·quiv·o·cate
[??kwiv??k?t]
VERB
equivocating (present participle)
use ambiguous language so as to conceal the truth or avoid committing oneself.

12. Huh?

So where do all these leave us with respect to “necessity”? It still seems “necessity is not a thing”.

107. 107
kairosfocus says:

NL, have you worked with dynamical analysis or system modelling using systems theory, differential equations or difference equations? KF

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kairosfocus says:

NL:

1: initial conditions are those convenient for a particular analysis, there is no necessary import of origin of cosmos or the like. (That is a commonplace of the analysis of dymamical systems — indeed in mechanics, “displacement” is distance moved in a relevant direction relative to a reference or initial point.)

2: sensitive dependence on INITIAL conditions is a core concept in chaos theory. Let me add a summary from Wikipedia, as a convenient source:

Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions. “Chaos” is an interdisciplinary theory stating that within the apparent randomness of chaotic complex systems, there are underlying patterns, constant feedback loops, repetition, self-similarity, fractals, self-organization, and reliance on programming at the initial point known as sensitive dependence on initial conditions. The butterfly effect describes how a small change in one state [–> the relevant initial state or intervening state] of a deterministic nonlinear system [–> notice, this is what mechanical necessity is about] can result in large differences in a later state, e.g. a butterfly flapping its wings in Brazil can cause a hurricane in Texas.[1]

Small differences in initial conditions, such as those due to rounding errors in numerical computation [–> or in a mechanical set-up etc, here we see how chance enters as a second causal factor], yield widely diverging outcomes for such dynamical systems, rendering long-term prediction [–> as opposed to sufficiently short term] of their behavior impossible in general.[2][3] This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved.[4] [–> apart from in the initial conditions] In other words, the deterministic nature of these systems does not make them predictable.[5][6] This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:[7]

Chaos: When the present determines the future, but the approximate present does not approximately determine the future.

Chaotic behavior exists in many natural systems, such as weather and climate.[8][9] It also occurs spontaneously in some systems with artificial components, such as road traffic . . . .

[Let’s add:] Sensitivity to initial conditions means that each point in a chaotic system is arbitrarily closely approximated by other points with significantly different future paths, or trajectories. Thus, an arbitrarily small change, or perturbation, of the current trajectory may lead to significantly different future behavior. [–> i.e. initial or intervening]

3: The singularity is indeed inferred from observed cosmological expansion and the like; this is similar to how the electron has never been directly observed. That is a commonplace of science, the issue is whether BB or electron are compelling, well supported best explanations for what we observe.

4: Darwin’s pond etc are indeed model scenarios, for argument possible worlds to be explored and analysed given that there is a debate of paradigms to be addressed.

5: Initial conditions are the baseline from which further interventions are considered. This, again is an analytical commonplace. Notice, we use one-sided Laplace transforms that start from a zero point, as do the classic test functions.

6: Similarly, precise predictive power is an important standard in science, engineering, modelling etc, that is where confidence in empirical reliability is built up.

7: We are discussing in a wider context where there are well known views that would explain reality on blind chance and mechanical necessity. From Plato on — this is history not blind appeal to authority [you knocked over a strawman] — it has been recognised on record that a third factor is relevant, intelligently directed configuration. A significant issue is to distinguish the three, and it is relevant to also show why the third does not plausibly reduce to the first two.

8: In your diagram, you try to equate necessity and design. You claim: “Design is just a set of ‘laws’, making the design-vs-necessity distinction impossible.” No, design is only possible where an intelligence can use purpose to impose a configuration. For DNA, the chaining of bases is contingent so information can be fed into particular sequencing. For the cosmos, the laws, constants etc are mathematical and can be manipulated through sensitivity analysis, which revealed fine tuning.

9: Kindly re-examine the analysis of logic of being. For one, compare the 2 x 2 options matrix: possible vs impossible of being with present in all worlds vs present in some. That sets a context of logical possibilities.

10: Now, ponder a fire and see that it has enabling factors, so it is contingent, it can begin or be blocked [that’s 2 neighbouring possible worlds] and can be sustained or cease or even be put out. This is how contingent beings are, telling us about cause.

11: What happens if a being — or better, a candidate being — is not dependent on enabling factors? It will either always be present in any world or else it will be impossible of being in any world. So far, a concept.

12: Is this serious? Yes, as we live in a contingent world, and non-being has no causal power, we need some things that always were and are independent of other being, as adequate root of such a world. If a world now is, SOMETHING always was of adequate capability for such a world to become actual, one that includes morally governed creatures — us.

13: It turns out we can see that a sufficient reason for such NB’s to be present in any world is that they are framework for a world to exist. Distinct identity and its consequences are examples: it is not possible for a world to be in which two-ness etc do not obtain.

And so forth.

KF

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kairosfocus says:

PS: Walker and Davies on dynamical systems:

In physics, particularly in statistical mechanics, we base many of our calculations on the assumption of metric transitivity, which asserts that a system’s trajectory will eventually [–> given “enough time and search resources”] explore the entirety of its state space – thus everything that is phys-ically possible will eventually happen. It should then be trivially true that one could choose an arbitrary “final state” (e.g., a living organism) and “explain” it by evolving the system backwards in time choosing an appropriate state at some ’start’ time t_0 (fine-tuning the initial state). In the case of a chaotic system the initial state must be specified to arbitrarily high precision. But this account amounts to no more than saying that the world is as it is because it was as it was, and our current narrative therefore scarcely constitutes an explanation in the true scientific sense.

We are left in a bit of a conundrum with respect to the problem of specifying the initial conditions necessary to explain our world. A key point is that if we require specialness in our initial state (such that we observe the current state of the world and not any other state) metric transitivity cannot hold true, as it blurs any dependency on initial conditions – that is, it makes little sense for us to single out any particular state as special by calling it the ’initial’ state. If we instead relax the assumption of metric transitivity (which seems more realistic for many real world physical systems – including life), then our phase space will consist of isolated pocket regions and it is not necessarily possible to get to any other physically possible state (see e.g. Fig. 1 for a cellular automata example).

[–> or, there may not be “enough” time and/or resources for the relevant exploration, i.e. we see the 500 – 1,000 bit complexity threshold at work vs 10^57 – 10^80 atoms with fast rxn rates at about 10^-13 to 10^-15 s leading to inability to explore more than a vanishingly small fraction on the gamut of Sol system or observed cosmos . . . the only actually, credibly observed cosmos]

Thus the initial state must be tuned to be in the region of phase space in which we find ourselves [–> notice, fine tuning], and there are regions of the configuration space our physical universe would be excluded from accessing, even if those states may be equally consistent and permissible under the microscopic laws of physics (starting from a different initial state). Thus according to the standard picture, we require special initial conditions to explain the complexity of the world, but also have a sense that we should not be on a particularly special trajectory to get here (or anywhere else) as it would be a sign of fine–tuning of the initial conditions. [ –> notice, the “loading”] Stated most simply, a potential problem with the way we currently formulate physics is that you can’t necessarily get everywhere from anywhere (see Walker [31] for discussion). [“The “Hard Problem” of Life,” June 23, 2016, a discussion by Sara Imari Walker and Paul C.W. Davies at Arxiv.]

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Nonlin.org says:

kf,

Again, I am amazed and slightly annoyed by your capacity to complicate simple matters. Sorry, I will have to jump to what matters and disregard the rest.

You say:
“there are well known views that would explain reality on blind chance and mechanical necessity. From Plato on — this is history not blind appeal to authority [you knocked over a strawman] — it has been recognised on record that a third factor is relevant, intelligently directed configuration. A significant issue is to distinguish the three, and it is relevant to also show why the third does not plausibly reduce to the first two.”

…and then proceed to make a very obscure and unconvincing case for “necessity different than design”.

In contrast, my case is very clear and simple as explained in par 5: “Necessity is Design to the best of our knowledge…”

For instance, you say: “For the cosmos, the laws, constants etc are mathematical and can be manipulated through sensitivity analysis, which revealed fine tuning.”

Are you saying “the laws of physics we observe are “mathematical” (universal and frozen for ever?). But this is not true as I explain in par 6: “Scientific laws are unknowable. Only instances of these laws are known with any certainty…”

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ET says:

Calm down. Breathe. Take it one point at a time and keep it brief. Please

(both of you)

It becomes an instant turn-off to any readers when you talk past each other and post much more than what needs to be said.

Merry Christmas

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kairosfocus says:

ET, it is obvious that further discussion will not be helpful as it takes two parties willing to engage the substance of a matter for such progress. I am closing this thread. KF