Do all mathematical proofs have value?

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Re: Do all mathematical proofs have value?

#21  Postby SafeAsMilk » Mar 31, 2018 11:08 am

LucidFlight wrote:
jamest wrote:Lots of 'proofs'. Do they ALL have metaphysical potential and hence value? That's what I want to discuss.

In order to help the discussion along, what would be an example of a proof with metaphysical potential, in your opinion? And, what do you mean by metaphysical potential?

I do wish jamest would answer this excellent post, which would set us on the path of discovering what he really wants to discuss.
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Re: Do all mathematical proofs have value?

#22  Postby felltoearth » Mar 31, 2018 1:30 pm

SafeAsMilk wrote:
LucidFlight wrote:
jamest wrote:Lots of 'proofs'. Do they ALL have metaphysical potential and hence value? That's what I want to discuss.

In order to help the discussion along, what would be an example of a proof with metaphysical potential, in your opinion? And, what do you mean by metaphysical potential?

I do wish jamest would answer this excellent post, which would set us on the path of discovering what he really wants to discuss.

It’s the sum of all stored metaphysicals waiting to be relaeased to the world. The process of transforming to kinetic metaphysicals however, means that the worthiness constant needs to be overcome by additional input from an entity which only exists in god’s mind. Thus we get Mp*Mk=Wb2 where b is the energy required of the metaphysicals to reach the temperature of the badger measured in joules.
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Re: Do all mathematical proofs have value?

#23  Postby The_Metatron » Mar 31, 2018 4:12 pm

Temperature measured in joules? That’s why we have proofs, to keep such abominations out of the human body of knowledge.


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Re: Do all mathematical proofs have value?

#24  Postby zoon » Mar 31, 2018 4:27 pm

jamest wrote:I'm not a mathematician, yet respect the intelligence and genius required to be a good one. However, from a philosophical perspective I'd like to explore the value of ALL mathematical proofs. That is, I'm not denying that some (if not most) are valuable. I'm just curious as to whether they're ALL valuable. I mean, I saw Cali's post here:

http://www.rationalskepticism.org/fun-g ... l#p2623602

Lots of 'proofs'. Do they ALL have metaphysical potential and hence value? That's what I want to discuss.

I suppose that collecting all kinds of theorems is somewhat like collecting scientific facts about, for example, exotic microorganisms or stars; they are all nuggets of information which might come in handy some day, and which meanwhile add to our model of the world and reinforce the correctness of its basic assumptions (unless they don't, which is when they become especially interesting). I think there's human value (which is all the metaphysical value we've got) in exploration which is largely for the fun of it. Like others here, I would be interested to know what jamest's criterion of metaphysical value may be, whether it's markedly different from the common or garden value of genetic survival which led to the evolution of inquisitiveness, cooperation and other human traits.
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Re: Do all mathematical proofs have value?

#25  Postby Calilasseia » Mar 31, 2018 4:29 pm

Let's see ... James wants to know what the value of a system that repeatedly and reliably demonstrates the consistency of theorems with the underlying axioms in a rigorous manner, from the standpoint of assertionism.

You can't make this up. Or rather, those of us possessing some respect for proper discourse can't make this up.
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Re: Do all mathematical proofs have value?

#26  Postby Calilasseia » Mar 31, 2018 4:57 pm

Meanwhile, to answer the actual question, mathematical proofs provide answers to specific, well-defined questions, for which no other process can provide an answer. An excellent example is the modern day rendering of Euclid's proof of the infinitude of the prime numbers.

We begin with some basic definitions. First:

Exactly divisible. Let two positive integers exist, a and b. Then a is exactly divisible by b, if a/b is itself a positive integer.

As a corollary, whenever a is exactly divisible by b, a can be written in the form k × b, where k is another positive integer. In algebra, we frequently use the juxtaposition shorthand kb for this, but this is merely a notational convention.

This leads to the corollary definition of:

Exact divisor. Given two positive integers a and b, b is an exact divisor of a, if a/b is itself a positive integer.

Next:

Prime number. A number p is a prime number, if and only if the sole exact divisors of p, are 1 and p itself. Any number a which has exact divisors other than 1 and a itself, is a composite number. so called because it can be composed by multiplying together those other exact divisors.

We now come to:

Theorem. The prime numbers form an infinite set.

Proof. We prove this result by contradiction. We first assert, by hypothesis, that the prime numbers constitute a finite set, and prove that this leads to a contradiction. This assertion consists of the statement that a set:

S = {p1, p2, p3, ... , pk}

containing k members, with each of the pi in the set being the prime numbers from 2 upwards consecutively, comprises the totality of the prime numbers.

We now construct the following number:

P = p1 × p2 × p3 × ... × pk

This is a composite number (by definition), whose exact divisors are the prime numbers in the set S.

We now ask the elementary question, what are the exact divisors of (P+1)?

P+1 cannot have p1 = 2 as a divisor, since P+1 is of the form 2K + 1, which is not exactly divisible by 2.

Likewise, P+1 cannot have p2 = 3 as a divisor, since P+1 is of the form 3K + 1, which is not exactly divisible by 3.

Likewise, P+1 cannot have p3 = 5 as a divisor, since P+1 is of the form 5K + 1, which is not exactly divisible by 5.

This continues all the way to ...

P+1 cannot have pk as a divisor, since P+1 is of the form 3pk + 1, which is not exactly divisible by pk.

We are therefore left with two choices. Either:

[1] P+1 itself is prime, in which case we have found a prime number not contained in the original set S, contradicting our original assertion;

[2] P+1 is a composite number, and can therefore be written in the form (P+1) = rs. But the same logic above applies to r and s; neither of these can be divisible by any of the pi in our set S above. If any one of r and s is prime, the result again follows. This procedure can be conducted recursively, until we find prime exact divisors of (P+1) that are not members of the set S.

The assertion that a finite set S as defined above is complete, is therefore refuted by contradiction, because for all such sets S, it is possible to construct a number, from the members of S, that yields a prime number that is not a member of S, and therefore it is impossible to construct a finite set of primes that is complete. Therefore, there exists an infinite number of prime numbers.

The proof is reliable, and rigorous. If anyone has a result from metaphysics matching this level of rigour and reliability, I'll be happy to learn of it. Until then, I think pure mathematics needs to take no notice of assertionists. :)
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Re: Do all mathematical proofs have value?

#27  Postby SafeAsMilk » Mar 31, 2018 5:02 pm

Yeah but the point isn't whether they're useful in the real, observable world. The important question for metaphysics is: do they give you a braingasm?
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Re: Do all mathematical proofs have value?

#28  Postby Calilasseia » Mar 31, 2018 5:13 pm

SafeAsMilk wrote:Yeah but the point isn't whether they're useful in the real, observable world. The important question for metaphysics is: do they give you a braingasm?


Much of the real, observable world gives me a braingasm. Therefore metaphysics is superfluous to requirements and irrelevant from that standpoint as far as I'm concerned. :D
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Re: Do all mathematical proofs have value?

#29  Postby felltoearth » Mar 31, 2018 5:13 pm

The_Metatron wrote:Temperature measured in joules? That’s why we have proofs, to keep such abominations out of the human body of knowledge.


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Re: Do all mathematical proofs have value?

#30  Postby SafeAsMilk » Mar 31, 2018 5:33 pm

Calilasseia wrote:
SafeAsMilk wrote:Yeah but the point isn't whether they're useful in the real, observable world. The important question for metaphysics is: do they give you a braingasm?


Much of the real, observable world gives me a braingasm. Therefore metaphysics is superfluous to requirements and irrelevant from that standpoint as far as I'm concerned. :D

No no, a metaphysical braingasm. Like when your brain realizes it's God for a second, then goes and shit talks to your other selves on the internet.
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Re: Do all mathematical proofs have value?

#31  Postby scott1328 » Mar 31, 2018 5:38 pm

Isn’t axiomatic, formal proof the acme of pure assertionism?

Mathematicians assert axioms, rules of inference, and definitions. That these axioms, definitions, and rules of inference and the theorems derived therefrom actually do model the real world is left as a matter for scientists to decide.

The value of a mathematical model, then, could be determined by how useful such a model is.
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Re: Do all mathematical proofs have value?

#32  Postby The_Metatron » Mar 31, 2018 6:23 pm

felltoearth wrote:
The_Metatron wrote:Temperature measured in joules? That’s why we have proofs, to keep such abominations out of the human body of knowledge.


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Oh, no! I thought that was on purpose.


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Re: Do all mathematical proofs have value?

#33  Postby Calilasseia » Mar 31, 2018 10:08 pm

scott1328 wrote:Isn’t axiomatic, formal proof the acme of pure assertionism?

Mathematicians assert axioms, rules of inference, and definitions. That these axioms, definitions, and rules of inference and the theorems derived therefrom actually do model the real world is left as a matter for scientists to decide.

The value of a mathematical model, then, could be determined by how useful such a model is.


Not quite. Because one of the tasks that is then opened up, within the world of pure mathematics, is to establish that any set of axioms generated isn't internally inconsistent. Only then, do they move on to the process of establishing that if a given set of axioms holds for any system, then various theorems derive as a consequence for those same systems. The entire process is conditional upon whether or not the axioms hold. That's where pure mathematics differs from assertionism - pure mathematicians don't regard their axioms as unconditionally and universally true, instead, they regard their axioms as the foundations for abstract structures to be investigated in more depth. Though Euclid was far ahead of the game than most on this matter - so far ahead, that it took something like 18 centuries for other pure mathematicians to work out what he was actually proposing, when he axiomatised geometry. Which they only worked out when, at long last, they had the idea of proposing alternative axioms, and testing to see if those alternative axioms led to a consistent system. Once that leap was finally made, the rest was, as they say, history.

In short, pure mathematicians don't regard their axioms as prescriptive rules applicable to the universe and its contents, instead, they regard their axioms as the foundation for abstract structures of interest, and remain open to the possibility of alternative structures, when a different set of axioms allows such structures to be possible through their own consistency.
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Re: Do all mathematical proofs have value?

#34  Postby OlivierK » Apr 01, 2018 12:06 am

Proofs are no more than shortcuts that allow reasoning, including other proofs, to start from somewhere other than axioms (on the grounds that the path from axioms to the result of the proof is known to exist). But any time you use a proof, you could just start from axioms instead. They're timesaving devices, and have the same metaphysical value as dishwashers or high-speed railways or corner stores, whatever that metaphysical value may be (I find "none" to be a satisfactory answer, but because it's metaphysics, YMMV).

Also, if we're to be able to compare the metaphysical value of various proofs, we'd need a measurement system and agreed units. I propose that the unit of measurement be "unicorn equivalents" (UE) where the rather lovely Euclid proof Cali gave above = 1UE. I eagerly await a rigorous calculation of the UE value of a proof of Pythagoras' Theorem.
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Re: Do all mathematical proofs have value?

#35  Postby Thommo » Apr 01, 2018 2:39 am

Most structures of interest in mathematical logic (areas like model theory, proof theory, computability theory etc.) in modern mathematical research do not prove their own consistency. The reasons for that are (and we get to mention them in their appropriate context for once) Gödel's incompleteness theorems, any formal axiomatic system that contains basic arithmetic and proves its axioms are consistent is inconsistent.

Olivier's post starts with the word "proof" when he means "theorem" (or lemma or corollary), by the way.

If anyone can tell me the correct "value" or "use" (outside of mathematics) of a key result in logic like the
Löwenheim–Skolem theorem I'd certainly love to see them do it. I used to use that thing all the time, but I'm buggered if I can see any connection from it to anything non mathematical.
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Re: Do all mathematical proofs have value?

#36  Postby scott1328 » Apr 01, 2018 3:20 am

Calilasseia wrote:
scott1328 wrote:Isn’t axiomatic, formal proof the acme of pure assertionism?

Mathematicians assert axioms, rules of inference, and definitions. That these axioms, definitions, and rules of inference and the theorems derived therefrom actually do model the real world is left as a matter for scientists to decide.

The value of a mathematical model, then, could be determined by how useful such a model is.


Not quite. Because one of the tasks that is then opened up, within the world of pure mathematics, is to establish that any set of axioms generated isn't internally inconsistent. Only then, do they move on to the process of establishing that if a given set of axioms holds for any system, then various theorems derive as a consequence for those same systems. The entire process is conditional upon whether or not the axioms hold. That's where pure mathematics differs from assertionism - pure mathematicians don't regard their axioms as unconditionally and universally true, instead, they regard their axioms as the foundations for abstract structures to be investigated in more depth. Though Euclid was far ahead of the game than most on this matter - so far ahead, that it took something like 18 centuries for other pure mathematicians to work out what he was actually proposing, when he axiomatised geometry. Which they only worked out when, at long last, they had the idea of proposing alternative axioms, and testing to see if those alternative axioms led to a consistent system. Once that leap was finally made, the rest was, as they say, history.

In short, pure mathematicians don't regard their axioms as prescriptive rules applicable to the universe and its contents, instead, they regard their axioms as the foundation for abstract structures of interest, and remain open to the possibility of alternative structures, when a different set of axioms allows such structures to be possible through their own consistency.


I think you are romanticizing Euclid. He certainly had not the slightest inkling of axioms and formal systems.
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Re: Do all mathematical proofs have value?

#37  Postby Calilasseia » Apr 01, 2018 3:49 am

Thommo wrote:Most structures of interest in mathematical logic (areas like model theory, proof theory, computability theory etc.) in modern mathematical research do not prove their own consistency. The reasons for that are (and we get to mention them in their appropriate context for once) Gödel's incompleteness theorems, any formal axiomatic system that contains basic arithmetic and proves its axioms are consistent is inconsistent.

Olivier's post starts with the word "proof" when he means "theorem" (or lemma or corollary), by the way.

If anyone can tell me the correct "value" or "use" (outside of mathematics) of a key result in logic like the
Löwenheim–Skolem theorem I'd certainly love to see them do it. I used to use that thing all the time, but I'm buggered if I can see any connection from it to anything non mathematical.


Actually, the correct result arising from Gödel's Incompleteness Theorem, is that any formal axiomatic system whose axioms possess at least the expressive power of the axioms of elementry number theory, cannot be simultaneously consistent and complete. Complete here meaning that all true propositions of the system are derivable within the system. If the axioms of such a system are consistent then the system is incomplete. If the system is complete, then the axioms are inconsistent.

scott1328 wrote:
Calilasseia wrote:
scott1328 wrote:Isn’t axiomatic, formal proof the acme of pure assertionism?

Mathematicians assert axioms, rules of inference, and definitions. That these axioms, definitions, and rules of inference and the theorems derived therefrom actually do model the real world is left as a matter for scientists to decide.

The value of a mathematical model, then, could be determined by how useful such a model is.


Not quite. Because one of the tasks that is then opened up, within the world of pure mathematics, is to establish that any set of axioms generated isn't internally inconsistent. Only then, do they move on to the process of establishing that if a given set of axioms holds for any system, then various theorems derive as a consequence for those same systems. The entire process is conditional upon whether or not the axioms hold. That's where pure mathematics differs from assertionism - pure mathematicians don't regard their axioms as unconditionally and universally true, instead, they regard their axioms as the foundations for abstract structures to be investigated in more depth. Though Euclid was far ahead of the game than most on this matter - so far ahead, that it took something like 18 centuries for other pure mathematicians to work out what he was actually proposing, when he axiomatised geometry. Which they only worked out when, at long last, they had the idea of proposing alternative axioms, and testing to see if those alternative axioms led to a consistent system. Once that leap was finally made, the rest was, as they say, history.

In short, pure mathematicians don't regard their axioms as prescriptive rules applicable to the universe and its contents, instead, they regard their axioms as the foundation for abstract structures of interest, and remain open to the possibility of alternative structures, when a different set of axioms allows such structures to be possible through their own consistency.


I think you are romanticizing Euclid. He certainly had not the slightest inkling of axioms and formal systems.


Despite the fact that his geometry is the prime example of such a system? Furthermore, Book 7 of The Elements deals with an axiomatisation of elementary number theory, though one that is different from modern versions thereof (such as those arising post-Peano).

To say that he had no inkling of axioms and formal systems, when he explicitly constructed two such systems in The Elements, strikes me as a remarkable assertion. He may not have had the modern understanding of axiomatic formal systems, but he certainly understood the value of laying down foundational principles from which other, more general results logically follow, which is a central part of developing such systems.
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Re: Do all mathematical proofs have value?

#38  Postby scott1328 » Apr 01, 2018 3:51 am

Have you actually read the elements?
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Re: Do all mathematical proofs have value?

#39  Postby Greyman » Apr 01, 2018 4:19 am

jamest wrote:
Thomas Eshuis wrote:Does metaphysics have value?

Of course metaphysics has value. That's why we're all blowing each other up with explosives, etc..
It is interesting to note that nobody was blowing each other up over the effort to prove Fermat's Last Theorem or... well, any mathematical proofs or conjectures, ever. The only argument that is of any worth is one of valid reasoning from justified premises.

I'd suggest that the need to resort to explosives is a good indication that the metaphysics doesn't have value.
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Re: Do all mathematical proofs have value?

#40  Postby Thommo » Apr 01, 2018 4:27 am

Calilasseia wrote:
Thommo wrote:Most structures of interest in mathematical logic (areas like model theory, proof theory, computability theory etc.) in modern mathematical research do not prove their own consistency. The reasons for that are (and we get to mention them in their appropriate context for once) Gödel's incompleteness theorems, any formal axiomatic system that contains basic arithmetic and proves its axioms are consistent is inconsistent.

Olivier's post starts with the word "proof" when he means "theorem" (or lemma or corollary), by the way.

If anyone can tell me the correct "value" or "use" (outside of mathematics) of a key result in logic like the
Löwenheim–Skolem theorem I'd certainly love to see them do it. I used to use that thing all the time, but I'm buggered if I can see any connection from it to anything non mathematical.


Actually, the correct result arising from Gödel's Incompleteness Theorem, is that any formal axiomatic system whose axioms possess at least the expressive power of the axioms of elementry number theory, cannot be simultaneously consistent and complete. Complete here meaning that all true propositions of the system are derivable within the system. If the axioms of such a system are consistent then the system is incomplete. If the system is complete, then the axioms are inconsistent.


That's not a correction at all.

What I said is 100% correct. Any formal system that embeds Peano arithmetic and proves its own consistency is inconsistent. If you don't believe me just look up the second incompleteness theorem on a wiki or something.
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