David Hilbert wanted all mathematics to be proved by logical steps. Kurt GĂśdel showed that no axiomatic system could be complete and consistent at the same time:
On Monday, September 8, 1930, Hilbert opened the annual meeting of the Society of German Scientists and Physicians in KĂśnigsberg with a famous discourse called âLogic and the knowledge of nature.â He ended with these words:
âFor the mathematician there is no Ignorabimus, and, in my opinion, not at all for natural science eitherâŚ
âThe true reason why [no-one] has succeeded in finding an unsolvable problem is, in my opinion, that there is no unsolvable problem. In contrast to the foolish Ignorabimus, our credo avers: We must know, We shall know.â
In one of those ironies of history, during the three days prior to the conference opened by Hilbertâs speech, a joint conference called Epistemology of the Exact Sciences also took place in KĂśnigsberg. On Saturday, September 6, in a twenty-minute talk, Kurt GĂśdel (1906â1978) presented his incompleteness theorems. On Sunday 7, at the roundtable closing the event, GĂśdel announced that it was possible to give examples of mathematical propositions that could not be proven in a formal mathematical axiomatic system even though they were true.
The result was shattering. GĂśdel showed the limitations of any formal axiomatic system in modeling basic arithmetic. He showed that no axiomatic system could be complete and consistent at the same time.
Daniel AndrĂŠs DĂaz PachĂłn, “Faith is the most fundamental of the mathematical tools” at Mind Matters News
So itâs not a question of faith vs. reason but faith so we can have reason.
Intelligent Design has no need for faith. The evidence is all around us. Mathematics is something to be discovered, since it already exists. There are absolutes in the universe, which could not have come about by chance. It is the Darwinists that must rely on faith, since there is no evidence to support their claim. They point to micro-evolution, but that’s not proof of macro-evolution. They point to fossils, but ignore the lack of mutations and millions of years worth of gaps. They point to all sorts of things, but never actual proof of their delusional beliefs.
Of course maths depends on unprovable assumptions, and this was known long before GĂśdel. That’s why they are assumptions not theorems or lemmas. The history of Euclid’s parallel postulate is a great example of this.
Godel was shattering to theoreticians but utterly inconsequential to people who actually use math. Those “foundations” are not foundations at all, only decorations tacked on after math was fully developed. Math developed without anyone “knowing” about those “foundations”, and math continues to develop without any reference at all to those “foundations”.
Bob O’H
Wrong. As the OP demonstrates, one of the most famous and influential mathematicians of the early 20th Century did not know this.
Alfred North Whitehead and Bertrand Russell did not know this when they wrote their Principia Mathematica, one of the most influential texts of the early 20th Century.
Bob, I thought you were a math teacher (maybe I am wrong about this). But if you are, it is truly astonishing that you could be so spectacularly wrong about the history of math.
Bob O’Hara has two very different faiths that are irreconcilable with each other. On the one hand Bob O’Hara has faith that all life on earth is the result of unguided and purposeless materialistic processes (,,i.e. Darwinian materialism writ large). In philosophical terms, Bob O’Hara is a reductive materialist who believes that all life and mind are reducible to materialistic explanations. On the other hand, Bob O’Hara has faith that mathematics is undeniably true. In fact, Bob O’Hara himself makes his living from mathematics as a statistician
And herein lies the irresolvable dilemma for Bob O’Hara, mathematics, (which he himself uses so as to try to prove that Darwinian evolution is somehow scientifically/mathematically feasible), is not reducible to any possible materialistic explanation.
That is to say, the existence of Mathematics itself is simply devastating to Bob’s Darwinian worldview since mathematics itself exists in a immaterial, beyond space and time, âPlatonic Realmâ, that simply is not reducible to any possible reductive materialistic explanation of Darwinian evolution.
As David Berlinski explains, âMathematicians are capable of grasping a world of objects that lies beyond space and timeâŚ.â
Simply put, Mathematics itself, contrary to the materialistic presuppositions of Darwinists, does not need the physical world in order to exist. As Dr. Michael Egnor notes, âMathematics is entirely about concepts, which have no precise instantiation in nature,,,â
And yet Darwinian materialists, although they deny that anything beyond the material realm exists, need this immaterial “Platonic realm” of mathematics in order for their theory to even be considered scientific in the first place. As M. Anthony Mills explains, âAnd yet â hereâs the rub â these âabstract (mathematical) objectsâ 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.â
The predicament that Darwinists 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 for their materialistic theory to even be considered scientific in the first place, should be the very definition of a scientifically self-refuting worldview.
Moreover, as should be obvious by now, the fact that man himself has access to, and can use, this transcendent, beyond space and time, immaterial world of mathematics, offers fairly compelling evidence that man in not a purely material being but that man must also possess a transcendent, beyond space and time, immaterial mind and/or soul.
As Charles Darwinâs contemporary, Alfred Russel Wallace himself stated, âNothing in evolution can account for the soul of man. The difference between man and the other animals is unbridgeable. Mathematics is alone sufficient to prove in man the possession of a faculty unexistent in other creatures. Then you have music and the artistic faculty. No, the soul was a separate creation.â
Thus in conclusion, we see that Bob OâHaraâs faith in, and use of, the immaterial âPlatonic realmâ of mathematics, in and of itself, refutes the faith that Bob has in his materialistic Darwinian worldview, and furthermore mathematics itself instead offers fairly compelling proof that he, statistician Bob OâHara himself, must possess an immaterial mind and/or soul.
I have a question for you Bob,
Verse:
Indeed, mathematics is based on the assumption that humans are capable of reasoning logically.
Inferring from the so-called rationally of our regular interlocutors there is a lot of evidence to the contrary.
I share all of BA77s concerns.
I’ll say also though that it’s good to know Professor O’Hara’s background. Knowing him as a real human being and not an anonymous commentator is a benefit.
My (and I’m sure “our”) opposition is not personal or intended to be disrespectful.
Barry @ 4 –
Really? if I’m wrong and you’re right, then every assumption in mathematics should be provable.
I’ll be generous and not going to ask that you show that every assumption is provable, I’ll let you off if you can show that the parallel postulate is provable from Euclid’s other postulates (or at least point to a valid proof of it). If you can’t do that, I’d suggest you retract your comment.
BTW,i it’s clear from p1 of the Principia Mathematica that Whitehead and Russell did know that they were making unprovable assumptions – they specifically say that they “are diminishing to the utmost the number of the undefined ideas and undemonstrated propositions”.
Here again, briefly, is a discussion I had with Ed George over a year ago (12/11/18) dealing with the question whether mathematics is a human invention or discovery. I think that question has a lot of relevance to what we are discussing here. After all, if itâs just a human invention as Ed believes and itâs based on unprovable assumptions then how reliable is mathematics?
I argued was that Edâs reasoning went like this:
To which Ed, apparently without embarrassment responded:
https://uncommondescent.com/mathematics/logic-first-principles-4-the-logic-of-being-causality-and-science/#comment-669576
In other words, Ed is âarguing,â âI donât know, therefore, nobody knows.â
But the question then is how does he know nobody else knows?
I would argue that Edâs position is self-refuting, therefore, itâs a non-starter. An argument which is only an argument about a personal opinion is not really an argument, itâs only being pointlessly argumentative. The objective of any logical argument is to establish the truth. Doubling down on oneâs personal beliefs doesnât move the ball in either direction. It is nothing more than a self-serving combination of hubris and dogmatism.
Are the unprovable assumptions underlying mathematics any different? If so how?
Silver Asiatic, you may appreciate this as well:
I think that argument fits hand in glove with Godel’s incompleteness theorems as well as with what was discussed previously in post 5 about the human mind necessarily being immaterial.
Bob:
What are you talking about? My comment went to your wildly inaccurate representation regarding history. It had nothing to do with math as such.
Barry, unless you’re going to claim that Euclid lived in the 20th century (!), I think you’ve utterly failed to understand my point. The statement the “Even mathematics depends on some unprovable assumptions” has been obvious for a long time, and has nothing to do with GĂśdel. It’s why Euclid had to come up with his postulates: these are the unprovable assumptions of Euclidean geometry. The parallel postulate is an interesting example, because mathematicians spent centuries worrying about it: they didn’t like it and there were attempts to prove it from the other postulates. In the 19th century they realised that they could replace it with an alternative postulates (which, of course, still couldn’t be proved) and so inventing non-Euclidean geometry.
So, historically, it’s clear that mathematicians knew that maths depends on unprovable assumptions, and this can be traced at least as far back as Euclid.
Bob,
Do you deny that Whitehead and Russell set out to demonstrate that every foundational principle of mathematics could be formally proved, starting with the proposition that 1+1=2, which they famously devoted 68 pages to proving?
The thing about famous factoids like “PN takes xx pages to proove 1 + 1 =2” is they are very often wrong. 68 pages seems to be your own misremembering of the factoid, but whatever number you put it is just as wrong. The proof might appear on page 379 or 1201, but it’s not like the whole book was leading to that point.
Moreover, as Bob says, Whitehead and Russell where trying to minimize the number of axioms require d and base mathmatical proofs in formal logic. Even then, they needed the axioms of choice and infinity.
Barry – Yes. Read what they wrote (see the link to the precise page on my comment 8). They specifically say that they âare diminishing to the utmost the number of the undefined ideas and undemonstrated propositionsâ. They don’t say they are eliminating them, just reducing them as far as they can. So their proof of 1+1=2 relies on some assumptions, and they specifically say on p15 “Some propositions must be asserted without proof, since all inference proceeds from propositions previously asserted”.
Whitehead & Russell prove 1+1=2 from a more primitive set of propositions (IIRC from set theory, but they probably had to establish set theory first).
At 2 Bob O’Hara states
And yet although Euclidâs parallel postulate was never proven it was, none-the-less, disproven::
Where this gets interesting is when we realize that Einstein’s Relativity, (both special and general relativity) themselves are not based upon 3-Dimensional Euclidean geometry but are instead based upon non-Euclidean geometry.
The result of these “non-Euclidean geometries” that undergird both theories of relativity is that spacetime itself curves, (and therefore Euclidâs parallel postulate will not hold as being true within the non-Euclidean geometry of the universe) .
And whereas the curvature of the 4-dimensional spacetime that undergirds Einstein’s general relativity is fairly well known,,,
,,, and whereas that non-Euclidean geometry associated with general relativity is fairly well known, what is far less well known is that special relativity also has spacetime curvature associated with it,
In the following video clip, which was made by two Australian University Physics Professors, we find that the 3-Dimensional world âfolds and collapsesâ into a tunnel shape as a âhypotheticalâ observer approaches the âhigher dimensionâ of the speed of light.
More interesting still, although the universe itself is based upon non-Euclidean geometry and therefore “has all sorts of deformations in space-time where it varies from the perfectly flat” none-the-less when we “average all those small-scale effects out and look at the big picture’ we find that, in spite of the universe itself being based upon non-Euclidean geometry, that “within an incredibly small margin of uncertainty, is that the universe is flat.,,,”. Moreover there are “no laws of physics that predict or restrict the topology.”
We simply should not be living in a ‘flat’ universe where parallel lines stay parallel. As John Gribbin pointed out, “”The Universe today is actually very close to the most unlikely state of all, absolute flatness.” And, “any deviation of the Universe from flatness in the Big Bang would have grown, and grown markedly, as the Universe expanded and aged. Like the pencil balanced on its point and given the tiniest nudges, the Universe soon shifts away from perfect flatness.”
In fact, in so far as measurement accuracy will allow, “astronomers estimate that the universe must have been flat to 1 part within 1Ă10^57 parts.”
Moreover, it is also important to point out that 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 âflatâ. As Fraser Cain also stated in the preceding article, âWe 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.,,,â
The author of the preceding article should be very grateful that the universe is âever-so-boringly flatâ. If the universe were not so âever-so-boringly flatâ science and technology simply would not be possible for humans in the first place.
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, all but impossible. As Michael Schirber explains, if any of the constants varied, (or the spacetime of the universe varied for that matter), then, “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.”
Moreover, this exceptional flatness for the universe allows us to see that the âtiny temperature variations (in the Cosmic BackGround Radiation) correspond to the largest scale structures of the observable universe.â
And what is interesting in regards to âthese largest scale structures of the observable universeâ, is that astronomy now reveals a surprising rotational coincidence for Earth in relation to the quasar and radio galaxy distributions in the universe, i.e âthese largest scale structures of the observable universeâ:
Moreover, when the tiny temperature variations are averaged, âsmearedâ, and/or smoothed out, they were able to detect the anomalies in the CMBR, which âstrangelyâ line up with the earth and solar system,
Here is an excellent clip from the documentary âThe Principleâ that explains, in an easy to understand manner, how these âanomaliesâ that line up with the earth and solar system were found, via âaveraging outâ, in the tiny temperature variations in the CMBR data.
In other words, the “tiny temperature variations” in the CMBR, to the largest scale structures in the universe itself, reveal teleology, (i.e. a goal directed purpose, a plan, a reason), that specifically included the earth from the start. ,,, The earth, from what our best science can now tell us, is not some random cosmic fluke as atheists had presupposed.
Moreover, whereas atheists have no clue why the universe should be exceptionally flat, (i.e. why parallel line should remain parallel), so as to allow man to ‘make sense’ of the universe in the first place. The bible predicted that the universe would be exceptionally flat thousands of years before it was discovered by modern astronomy:
As both Einstein and Wigner pointed out, the fact that mathematics should even be applicable to our universe in the first place, is, by all rights, a “miracle” that is inexplicable for atheists:
Supplemental notes:
Verse:
ba77 @ 16 –
No it wasn’t. It was shown that it’s possible to construct geometries where the parallel postulate does not hold. In Euclidean geometry, the parallel postulate is true (by assumption), so can’t be dis-proven.
BA77 @ 10
I had not seen that before – yes, it’s great.
There are similar concepts to that also.
Here, we have truths which are eternal.
It could also be universals which would exist even if there was no human life on earth.
Or immaterial forms of objects. Like a triangle.
Everyone who knows what a triangle is, can think of a triangle.
This triangle in thought, is the form – it is not a physical object, but it conforms to the geometric formula for a triangle.
1. Form or pattern of individual objects exist (the form of a triangle)
2. The form is a universal, and not a physical object but a mental object.
3. The form would exist even if there were no human beings on earth.
4. So, the form cannot be the product of human minds.
5. Therefore the form, essence or pattern of the object is immaterial and the product of a transcendent, universal mind.
Bob O’Hara, for crying out loud, I did say that it did not hold by assumption for euclidean geometry. My point was exactly that it was proven not to hold for non-euclidean geometries, i.e. The actual non-euclidean geometries which describe, via relativity, our universe! I went on to show, via the 1 in 10^57 fine tuned flatness of the universe, how atheists, such as you yourself, have no clue why euclidean geometry should even hold for our universe and thus for why humans should even be able to ‘miraculously’ describe the universe with mathematics, nor any clue for why we can apply mathematics in such a way so as to intelligently design scientific instruments so as to be able to even ‘do science’ in the first place.
The Theistic implications of the entire situation, which I made clear in posts 16 and 17, are overwhelming.
Missing the forest for the trees is too subtle a rebuke for your mistake.
Silver Asiatic at 19, thanks for fleshing that out. I like the fact that you used a triangle to flesh it out. (A triangle happens to be the same exact example I used in conversation with a friend yesterday to prove to him that his thoughts must be ‘spiritual’ instead of material)
correction, “Bob OâHara, for crying out loud, I did NOT say that it did not hold by assumption for euclidean geometry. ,,,,”
ba77 –
Well, perhaps you should have written that, then.
Bob (and weave) O’Hara states..
Willful blindness?
post 17
Your phrasing is somewhat unlucky: It was not proven that Euclid’s fifths postulate does not hold for non-euclidean geometries – that is obvious. It was shown that the fifths postulate is independent of the other four, the same way as the continuum hypothesis is independent from ZFC. So, you can add it (or some alternate version) to your postulates and see what kind of mathematics turns out….
My point is simply that the Parallel Postulate does not hold for the actual (real world) non-euclidean geometries (General and Special Relativity) which describe our universe! And that atheists have no clue, via the 1 in 10^57 fine-tuned flatness within that non-euclidean geometry, why that should be so. (see posts 16 and 17)
Whereas the bible predicted that the universe would be exceptionally flat thousands of years before it was discovered by modern astronomy:
Once again, if that was your point, you should have written that.
Bob: “Once again, if that was your point, you should have written that.”
I did write it over and over again, It is not my fault that you either ignored it, are that you have a reading comprehension issue.
To repeat:
@BA77: In your own words
This is followed by a quotation which does not make this point.
This lead me to the following observations:
1) you believe that the parallel postulate was disproven.
2) your quotations are only loosely connected to your statements.
ba77 –
This is what you write that I was commenting on:
Post 16:
Which does not say that the universe is almost flat.
Post 20:
Again, it does not say that the universe is almost flat.
Nobody else was discussing physics here, so it was a puzzle why you were changing the subject: it simply wasn’t relevant to whether the parallel postulate had been disproved, or if maths depends on unprovable assumptions. Perhaps if you had made it clear that you wanted to change the subject, and you weren’t discussing the development of non-Euclidean geometry (by, for example, not commenting on the development of non-Euclidean geometry) things might have been clearer.
Bob (and weave) O’Hara
You have heard of context have you not? The CONTEXT of my post is unambiguous. That you failed to take it into consideration is, again, not my fault.
Nor does your frivolous objection that ignores context even begin to answer the fact that atheists are a completely loss to explain the 1 in 10^57 fine-tuned flatness of the universe in a ‘non-euclidean’ universe.
I am more than satisfied that the unbiased reader can clearly see that the atheist is, once again, at a complete loss to explain why the universe is fine-tuned such as it is.
Whatever DiEb, the simple point remains that the parallel postulate does not hold as being ‘always’ true for the ‘non-euclidean’ (real world) universe we live in. As a result, atheists are a completely loss to explain the 1 in 10^57 fine-tuned flatness of the universe in a ânon-euclideanâ universe.
If you, as an atheist, want to instead argue that the parallel postulate is never-the-less true in some “Platonic realm” of mathematics, I will simply refer you to post 5 and point out that you, as a Darwinian materialist, have no right to the immaterial “Platonic realm” of mathematics in the first place:
Thus DiEb you lose on two fronts. The parallel postulate does not hold as ‘always’ being true in the ‘real world’ ‘material’ universe, and you have no right, as a Darwinian materialist, to the immaterial Platonic realm.
Choose your poison, both are fatal to you as a Darwinian materialist.
I’m curious: in what context does the sentence “And yet although Euclidâs parallel postulate was never proven it was, none-the-less, disproven” mean that Euclid’s parallel postulate was not disproven?
@BA77: “whatever”? As in “What ever I said before was wrong, but here are two new, irrelevant factoids?”
Split hairs much DiEb. I clearly was talking about the ‘real world’ of the non-euclidean geometry of this universe when I said the parallel postulate did not hold as being ‘always’ true and therefore was ‘disproven’. i.e. Disproven in so far as empirical science itself in concerned. You might ought to give up Darwinian evolution and try empirical science someday.
You want to quibble that it was not mathematically ‘disproven’. I say, “So what?” If your mathematical model cannot hold for the real world then it is a completely useless mathematical fantasy that has no bearing on the real world as far as I am concerned!
You say that the fact that the parallel postulate does not hold for the real world, and the fact that you, as a Darwinian materialist, have no right to the immaterial Platonic realm of mathematics in the first place are quote unquote “irrelevant factoids”.
HA!
It is not surprising that you would deny that two devastating facts to your materialistic worldview are relevant. It is par for the course. Nonetheless, I am more than satisfied that the unbiased reader can clearly see who is being forthright and who who is being disingenuous,
bs77 @ 31 –
Yes. hte contexct of your posts has been whether maths depends on unprovable assumptions, and specifically about the parallel postulate and whether it was an assumption or could be proved. In that context, it’s difficult to see why you change the subject to relativity.
Whatever Bob (and weave). You and Dieb can talk to each other and comfort yourselves in your own delusions as far as I care. My posts in 5, 16 and 17 speak for themselves and both arguments are both unscathed by your counter arguments.
Again, I’ll let the unbiased readers decide for themselves who is being forthright.
@BA77
Great idea! I suppose this unbiased reader will be impressed by your oh so clever piece of repetitive humor, too: bob and weave you have me in stitches e v e r y s i n g l e t i m e !
DiEb,
I’ve been an unbiased reader here for awhile and watching someone engaging Bob O’H invariably results in exactly what BA77 characterizes, which is B O’H Bobbing and Weaving, Leaving and then Returning to Bob and Weave some more.
Andrew
Andrew
And I am Mary, Queen of Scots. đ
I am Groot
ET
Not likely. Everybody loves Groot. đ
I. AM. GROOT.
“And I am Mary, Queen of Scots”
Ed George,
In an infinite number of universes you are. đ
And I’m unbiased by sheer weight of numbers. The amount of universes in which I’m unbiased is infinite. Mathematically speaking, I’m the Primus Unbias. đ
Andrew
Well, at least none of you is Spartacus.