Faith: Even mathematics depends on some unprovable assumptions

_{Denyse O'Leary

January 5, 2020

Intelligent Design, Mathematics}

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David Hilbert wanted all mathematics to be proved by logical steps. Kurt Gödel 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 Königsberg with a famous discourse called “Logic and the knowledge of nature.” He ended with these words:
“For the mathematician there is no Ignorabimus, and, in my opinion, not at all for natural science either…
“The 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.”
In one of those ironies of history, during the three days prior to the conference opened by Hilbert’s speech, a joint conference called Epistemology of the Exact Sciences also took place in Königsberg. On Saturday, September 6, in a twenty-minute talk, Kurt Gödel (1906–1978) presented his incompleteness theorems. On Sunday 7, at the roundtable closing the event, Gödel 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. Gödel 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 Andrés Díaz Pachón, “Faith is the most fundamental of the mathematical tools” at Mind Matters News

So it’s not a question of faith vs. reason but faith so we can have reason.

Comments

Barry - Yes. Read what they wrote (see the link to the precise page on my comment 8). They specifically say that they “are diminishing to the utmost the number of the undefined ideas and undemonstrated propositions”. 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).Bob O'H_{January 7, 2020
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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.Mimus_{January 6, 2020
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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?Barry Arrington_{January 6, 2020
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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 Gödel. 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 O'H_{January 6, 2020
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Bob:
Really? if I’m wrong and you’re right, then every assumption in mathematics should be provable.
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 Arrington_{January 6, 2020
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Silver Asiatic, you may appreciate this as well:
11. The Argument from Truth Excerpt: 1. Our limited minds can discover eternal truths about being. 2. Truth properly resides in a mind. 3. But the human mind is not eternal. 4. Therefore there must exist an eternal mind in which these truths reside. https://www.peterkreeft.com/topics-more/20_arguments-gods-existence.htm#11
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.bornagain77_{January 6, 2020
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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 it’s just a human invention as Ed believes and it’s based on unprovable assumptions then how reliable is mathematics? I argued was that Ed’s reasoning went like this:
Either X or Y could be true EG believes Y Therefore, Y is true. In other words, Ed George believes it. That settles it.
To which Ed, apparently without embarrassment responded:
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?
https://uncommondescent.com/mathematics/logic-first-principles-4-the-logic-of-being-causality-and-science/#comment-669576 In other words, Ed is ‘arguing,’ “I don’t know, therefore, nobody knows.” But the question then is how does he know nobody else knows? I would argue that Ed’s position is self-refuting, therefore, it’s a non-starter. An argument which is only an argument about a personal opinion is not really an argument, it’s only being pointlessly argumentative. The objective of any logical argument is to establish the truth. Doubling down on one’s personal beliefs doesn’t 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?john_a_designer_{January 6, 2020
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Barry @ 4 -
Of course maths depends on unprovable assumptions, and this was known long before Gödel.
Wrong. As the OP demonstrates, one of the most famous and influential mathematicians of the early 20th Century did not know this.
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".Bob O'H_{January 6, 2020
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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.Silver Asiatic_{January 6, 2020
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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.john_a_designer_{January 6, 2020
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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
Robert Brian O'Hara Professor - Department of Mathematical Sciences Excerpt: "Most of my work has been on ecology and evolution, using statistical methods to put data and models together to try to learn more about the real world." https://www.ntnu.edu/employees/bob.ohara
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, “Platonic Realm”, that simply is not reducible to any possible reductive materialistic explanation of Darwinian evolution.
Platonic mathematical world - image http://abyss.uoregon.edu/~js/images/platonic_physical.gif
As David Berlinski explains, “Mathematicians are capable of grasping a world of objects that lies beyond space and time….”
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/2013/10/found-upon-web-and-reprinted-here.html
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, “Mathematics is entirely about concepts, which have no precise instantiation in nature,,,”
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”?,,, 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,,, Furthermore, the very framework of Clark’s argument — logic — is neither material nor natural. Logic, after all, doesn’t exist “in the space-time continuum” and isn’t described by physics. What is the location of modus ponens? How much does Gödel’s incompleteness theorem weigh? What is the physics of non-contradiction? How many millimeters long is Clark’s argument for naturalism? Ironically the very logic that Clark employs to argue for naturalism is outside of any naturalistic frame. The strength of Clark’s defense of naturalism is that it is an attempt to present naturalism’s tenets clearly and logically. That is its weakness as well, because it exposes naturalism to scrutiny, and naturalism cannot withstand even minimal scrutiny. Even to define naturalism is to refute it. https://evolutionnews.org/2018/01/naturalism-and-self-refutation/
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, “And yet — here’s the rub — these “abstract (mathematical) 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.”
What Does It Mean to Say That Science & Religion Conflict? - M. Anthony Mills - April 16, 2018 Excerpt: Barr rightly observes that scientific atheists often unwittingly assume not just metaphysical naturalism but an even more controversial philosophical position: reductive materialism, which says all that exists is or is reducible to the material constituents postulated by our most fundamental physical theories. As Barr points out, this implies not only that God does not exist — because God is not material — but that you do not exist. For you are not a material constituent postulated by any of our most fundamental physical theories; at best, you are an aggregate of those constituents, arranged in a particular way. Not just you, but tables, chairs, countries, countrymen, symphonies, jokes, legal contracts, moral judgments, and acts of courage or cowardice — all of these must be fully explicable in terms of those more fundamental, material constituents. 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
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.
Dr. Ed Feser - The Immateriality of the Intellect - video Excerpt: 1: Formal thought processes can have an exact or unambiguous conceptual content. However, 2: Nothing material can have an exact or unambiguous conceptual content. So, 3: Formal thought processes are not material. https://www.youtube.com/watch?v=fNi0j19ZSpo
As Charles Darwin’s contemporary, Alfred Russel Wallace himself stated, “Nothing 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.”
“Nothing 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.” Alfred Russel Wallace – 1910 https://evolutionnews.org/2010/08/alfred_russel_wallace_co-disco/
Thus in conclusion, we see that Bob O’Hara’s faith in, and use of, the immaterial “Platonic realm” 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 O’Hara himself, must possess an immaterial mind and/or soul. I have a question for you Bob, Verse:
Mark 8:37 Is anything worth more than your soul?
bornagain77_{January 6, 2020
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Bob O'H
Of course maths depends on unprovable assumptions, and this was known long before Gödel.
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.Barry Arrington_{January 6, 2020
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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".polistra_{January 6, 2020
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Of course maths depends on unprovable assumptions, and this was known long before Gödel. That's why they are assumptions not theorems or lemmas. The history of Euclid's parallel postulate is a great example of this.Bob O'H_{January 6, 2020
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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.BobRyan_{January 5, 2020
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