What is a good proof in math?

What is the basis for reason? And mathematics?

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Philosophy Explorer
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What is a good proof in math?

Post by Philosophy Explorer »

There are certain proofs that are widely accepted while there are others that have limited acceptance to mathematicians. What are your criteria for a good proof? (for me it would be consistency)

PhilX
dionisos
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Re: What is a good proof in math?

Post by dionisos »

Philosophy Explorer wrote:There are certain proofs that are widely accepted while there are others that have limited acceptance to mathematicians. What are your criteria for a good proof? (for me it would be consistency)

PhilX
Without consistency, i would not consider it a proof at all, nor even a clue.

For me a good proof is a elegant proof, i mean by that, a proof as simple as possible, and which go directly to the substance of what is considered.
But i would accept a very inelegant proof.

Now i would be much more convinced by a proof that only use constructive reasoning.

Could you give a example of a mathematical proof with limited acceptance ?
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Re: What is a good proof in math?

Post by Philosophy Explorer »

dionisos wrote:
Philosophy Explorer wrote:There are certain proofs that are widely accepted while there are others that have limited acceptance to mathematicians. What are your criteria for a good proof? (for me it would be consistency)

PhilX
Without consistency, i would not consider it a proof at all, nor even a clue.

For me a good proof is a elegant proof, i mean by that, a proof as simple as possible, and which go directly to the substance of what is considered.
But i would accept a very inelegant proof.

Now i would be much more convinced by a proof that only use constructive reasoning.

Could you give a example of a mathematical proof with limited acceptance ?
I copied the following from Wiki which answers your question in part:

"To an intuitionist, the claim that an object with certain properties exists is a claim that an object with those properties can be constructed. Any mathematical object is considered to be a product of a construction of a mind, and therefore, the existence of an object is equivalent to the possibility of its construction. This contrasts with the classical approach, which states that the existence of an entity can be proved by refuting its non-existence. For the intuitionist, this is not valid; the refutation of the non-existence does not mean that it is possible to find a construction for the putative object, as is required in order to assert its existence. Existence is construction, not proof of non-existence (Fenstad). As such, intuitionism is a variety of mathematical constructivism; but it is not the only kind.

The interpretation of negation is different in intuitionist logic than in classical logic. In classical logic, the negation of a statement asserts that the statement is false; to an intuitionist, it means the statement is refutable[1] (e.g., that there is a counterexample). There is thus an asymmetry between a positive and negative statement in intuitionism. If a statement P is provable, then it is certainly impossible to prove that there is no proof of P. But even if it can be shown that no disproof of P is possible, we cannot conclude from this absence that there is a proof of P. Thus P is a stronger statement than not-not-P.

Similarly, to assert that A or B holds, to an intuitionist, is to claim that either A or B can be proved. In particular, the law of excluded middle, "A or not A", is not accepted as a valid principle. For example, if A is some mathematical statement that an intuitionist has not yet proved or disproved, then that intuitionist will not assert the truth of "A or not A". However, the intuitionist will accept that "A and not A" cannot be true. Thus the connectives "and" and "or" of intuitionistic logic do not satisfy de Morgan's laws as they do in classical logic.

Intuitionistic logic substitutes constructability for abstract truth and is associated with a transition from the proof to model theory of abstract truth in modern mathematics. The logical calculus preserves justification, rather than truth, across transformations yielding derived propositions. It has been taken as giving philosophical support to several schools of philosophy, most notably the Anti-realism of Michael Dummett. Thus, contrary to the first impression its name might convey, and as realized in specific approaches and disciplines (e.g. Fuzzy Sets and Systems), intuitionist mathematics is more rigorous than conventionally founded mathematics, where, ironically, the foundational elements which Intuitionism attempts to construct/refute/refound are taken as intuitively given."

PhilX
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Re: What is a good proof in math?

Post by Philosophy Explorer »

I also copied this from Wiki:

"According to Weyl 1946, 'Brouwer made it clear, as I think beyond any doubt, that there is no evidence supporting the belief in the existential character of the totality of all natural numbers ... the sequence of numbers which grows beyond any stage already reached by passing to the next number, is a manifold of possibilities open towards infinity; it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties, including the antinomies – a source of more fundamental nature than Russell's vicious circle principle indicated. Brouwer opened our eyes and made us see how far classical mathematics, nourished by a belief in the 'absolute' that transcends all human possibilities of realization, goes beyond such statements as can claim real meaning and truth founded on evidence. (Kleene (1952): Introduction to Metamathematics, p. 48-49)"
"Finitism is an extreme version of Intuitionism that rejects the idea of potential infinity. According to Finitism, a mathematical object does not exist unless it can be constructed from the natural numbers in a finite number of steps."

PhilX
dionisos
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Re: What is a good proof in math?

Post by dionisos »

Ok, like i had guessed, this is not really acceptance about proofs, but about axioms/principles.
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Re: What is a good proof in math?

Post by Philosophy Explorer »

Here is a specific example from Wiki about what an ultra-finitist thinks:

"I have seen some ultrafinitists go so far as to challenge the existence of 2^100 as a natural number, in the sense of there being a series of “points” of that length. There is the obvious “draw the line” objection, asking where in 2^1, 2^2, 2^3, … , 2^100 do we stop having “Platonistic reality”? Here this … is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas. I raised just this objection with the (extreme) ultrafinitist Yessenin-Volpin during a lecture of his. He asked me to be more specific. I then proceeded to start with 2^1 and asked him whether this is “real” or something to that effect. He virtually immediately said yes. Then I asked about 2^2, and he again said yes, but with a perceptible delay. Then 2^3, and yes, but with more delay. This continued for a couple of more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take 2^100 times as long to answer yes to 2^100 then he would to answering 2^1. There is no way that I could get very far with this.

Harvey M. Friedman 'Philosophical Problems in Logic'"
dionisos
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Re: What is a good proof in math?

Post by dionisos »

Yes, the less axioms you accept, the more certain you are of your proofs.

The ultra-finitist take this so far, that they can’t demonstrate a lot, but then their proofs are the more widely accepted proofs.

But constructivism seem largely enough for me.
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Re: What is a good proof in math?

Post by Philosophy Explorer »

Let me ask this secondary question: is infinity the only issue that splits up mathematicians into different camps of math ideology?

PhilX
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Re: What is a good proof in math?

Post by dionisos »

Philosophy Explorer wrote:Let me ask this secondary question: is infinity the only issue that splits up mathematicians into different camps of math ideology?

PhilX
No, but it is indeed a big thing, and most disagreement i see was on it.
To give you a example not linked to infinity, there are disagreement between the bayesian and frequentist view of probability.
Keep in mind that there are different kind of infinities, constructivist accept some kind of infinities. (not the ultra-finitist, but really this view is extreme)
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