The search for truth in mathematics

[PseuTonym]

However, the integers are defined as members of a set that satisfies a list of axioms.

The integers have been studied for thousands of years, and no particular attempt to define them is going to be universally accepted. You described one general approach to attempting to define the integers. Another approach is to not admit sets into the theory, and to have the non-negative integers as the values of the variables.

For example, according to Samuel Buss, there is a theory known as Robinson’s theory Q, introduced by Tarski, Mostowski, and Robinson in a 1953 publication. You can see for yourself how Q is defined (according to Buss) if you look at page 82 below:

“First-Order Proof Theory of Arithmetic.”
in Handbook of Proof Theory, edited by S. R. Buss.
Elsevier, Amsterdam, 1998, pp 79-147
Link:
math.ucsd.edu/~sbuss/ResearchWeb/handbookII/

(Note that the document consists of pages 79 to 147, so page 82 is actually the fourth page of the document. Also note that there is an error in the first axiom as given by Buss. He used a non-equal symbol, but he also used a negation symbol, with the effect of cancelling each other out. In fact, Sx is simply what we think of as x+1, and the range of values for the variables is the set of non-negative integers, so the axiom is supposed to say that there exists no x such that Sx = 0.)

It’s just a problem of translation. If I translate “integers” to whatever form that term would take within your axiomatic system, we would find our views are consistent.

Such terms do not exist. Q is simply a theory of the non-negative integers. I would say that the terminology “weak fragment of arithmetic” is potentially misleading. Q is very reliable. It is weak only in the sense that we are very restricted insofar as what we can deduce within Q. However, the effect is to protect us against some unpleasant surprises in future. Whatever problems might arise in future in set theory, Q seems to be immune. Can you explain how it would be possible to define within Q some fragment of arithmetic that includes all of the axioms of Q plus additional axioms?

 

[Oreoracle]

As interesting as math is, I think it’s overkill to delve into all of this just to evaluate the point I was making. My point is very simple, and it isn’t even a mathematical argument. It’s just an observation about language.

For example, suppose I define an unmarried man to be a “cat” rather than the more conventional “bachelor”. Both of us see a man with no ring on his finger. You say he is a bachelor, and I say he is a cat. It would appear to someone not savvy to our definitions that we disagree.

Now here is my question: In that scenario, would we actually disagree on anything more substantial than a convention; namely, what an unmarried man should be called? We both agree that the man is unmarried, so it seems to me that there is no real disagreement to speak of, just an issue of translation.

 

[PseuTonym]

As interesting as math is, I think it’s overkill to delve into all of this just to evaluate the point I was making.

You made more than one claim.

My point is very simple, and it isn’t even a mathematical argument. It’s just an observation about language.

I had the impression that you had attempted to apply something (perhaps an observation about language) to reach conclusions about mathematics. If not, then why is your observation about language in this thread?

Earlier, you wrote:

Thankfully, there’s a subtle point that should put any concerns you have to ease.

Is your point both simple and subtle?

For example, suppose I define an unmarried man to be a “cat” rather than the more conventional “bachelor”. Both of us see a man with no ring on his finger. You say he is a bachelor, and I say he is a cat. It would appear to someone not savvy to our definitions that we disagree.

Your scenario involving observation of a man not wearing a ring is a very good scenario because it illustrates something important. Disputes are typically about the underlying reality. In your scenario, there is no dispute about what has been observed. However, in your scenario, I don’t think that what has been observed is enough to reach a conclusion about the underlying reality.

There are facts in mathematics. When the authority of great mathematicians bumps up against facts, the mathematicians lose. For example …

According to John J. Watkins (Number Theory: A Historical Approach) …

In 1769, Euler made a conjecture that was as bold as Fermat’s famous conjecture that is known as “Fermat’s Last Theorem.” Euler claimed that no nth power could ever be written as a sum of fewer than n other nth powers. In other words, a cube would always require three cubes, a fourth power would require four fourth powers, and so on.

This conjecture lasted for almost two hundred years, but in 1968, Lander and Parkin discovered a counterexample:
(144 to the power of 5) = the sum of the 5th powers of the following numbers: 27, 84, 110, 133.

---

Now, it is possible for somebody to claim that the above counterexample is not a fact about any underlying reality, and that it is just a byproduct of language.

I don’t agree, but it is possible for somebody to make that claim. Do you make that claim?

 

[PseuTonym]

That is an excellent question, but the story provides no answer. I didn’t specify what The Emperor’s Axioms are. As far as I know, Gödel was working within standard meta-theory. I don’t think that Gödel introduced any new assumptions, except perhaps for omega-consistency. However, Rosser showed how the weaker assumption of mere consistency can be used to obtain Gödel’s conclusions. The theorems are still known as Gödel’s theorems, but there has been progress since the original publication.

The above revision seems to be correct. I have no idea what the source of my original statement was. It may have been simply my own confusion.

 

[Oreoracle]

I had the impression that you had attempted to apply something (perhaps an observation about language) to reach conclusions about mathematics. If not, then why is your observation about language in this thread?

Language, at least symbolic language, is more fundamental than mathematics.

Is your point both simple and subtle?

My point is that your different uses of “integers” amounts to equivocation. It’s simple in that it’s easy to understand how the equivocation happens. It’s subtle in that equivocation can be hard to notice if one isn’t careful.

Going back to the analogy of the bachelor, suppose I dislike having different terms for males and females who are unmarried, so I define “bachelor” to mean “unmarried adult” rather than having a separate term like “bachelorette”. Suppose you and I observe an unmarried woman. I claim she is a bachelor, and you may very well say she isn’t if you are using the more conventional definition.

So once again I ask: Would we disagree on anything more substantial than a convention in this case?

If you concede that we disagree only on the labels we are using, I would argue the same applies for different axiomatizations of “integers”. You’re using the same word to refer to different things.

 

[PseuTonym]

My point is that your different uses of “integers” amounts to equivocation. It’s simple in that it’s easy to understand how the equivocation happens. It’s subtle in that equivocation can be hard to notice if one isn’t careful.

I do not agree that I am using the word “integers” in different and conflicting ways. However, I can understand how you might reach that conclusion if you think of commitment to a list of axioms as being equivalent to merely acceptance of some definition of some word.

Perhaps we should begin by checking that we both mean the same thing by the word “equivocation.”

It would be an equivocation for somebody to claim that Euler’s conjecture was correct and to try to support that claim by denying that 144 to the power of 5 is an integer.

Suppose that Frege had invented a proof of Euler’s conjecture based on an inconsistent system of axioms. Suppose that Frege had lived to 1968 and had been notified of the counterexample, and had responded by saying the following:

“When I say that a statement in number theory is true, I simply mean that it can be deduced from my axioms. Thus, when I announced that Euler’s conjecture is true, I was not making a mistake.”

In that case, Frege would have been equivocating.

I must add that such a statement by Frege is purely hypothetical, and would be very much out of character for Frege.

 

[Oreoracle]

Perhaps we should begin by checking that we both mean the same thing by the word “equivocation.”

It would be an equivocation for somebody to claim that Euler’s conjecture was correct and to try to support that claim by denying that 144 to the power of 5 is an integer.

Suppose that Frege had invented a proof of Euler’s conjecture based on an inconsistent system of axioms. Suppose that Frege had lived to 1968 and had been notified of the counterexample, and had responded by saying the following:

“When I say that a statement in number theory is true, I simply mean that it can be deduced from my axioms. Thus, when I announced that Euler’s conjecture is true, I was not making a mistake.”

In that case, Frege would have been equivocating.

I must add that such a statement by Frege is purely hypothetical, and would be very much out of character for Frege.

In mathematics, agreement on definitions precedes all other discussion. There is no proof or conjecture to discuss before that. So if Frege tried to construct a proof, several things would already be “fixed”. The axiomatization of number theory he used would be fixed. The formulation of Euler’s conjecture would be fixed, and so forth.

Note that when you make a conjecture, you are making that conjecture with respect to a certain set of axioms. For example, if I conjectured that for any line L and any point not on that line P, there are infinitely many lines passing through P but not intersecting L, then my conjecture is false in Euclidean geometry but true in hyperbolic geometry. You cannot remove the conjecture from the set of axioms in which the conjecture was made.

So no, you can’t prove someone’s conjecture with respect to one system using a different system, because it isn’t the same conjecture.

 

[PseuTonym]

Note that when you make a conjecture, you are making that conjecture with respect to a certain set of axioms.

You are free to define the word “conjecture” as you like, but you should be aware that it is a very non-standard definition.

For example, the twin primes conjecture in its current form does not specify what axioms may be used.

According to the following video …

logic.harvard.edu/video.php?v=EFI_Magidor

… Wiles relied upon the assumption that there exist large cardinals. In other words, Wiles assumed not just ZFC set theory, but additional set theoretic assumptions.

However, you are unlikely to find any formulation of Fermat’s Last Theorem (the conjecture) that refers to axioms of set theory. After all, it is a conjecture of number theory, not set theory.

If a counter-example to Fermat’s Last Theorem is discovered, then it will be accepted that Fermat’s Last Theorem is false. That will be true even if Wiles’ reasoning is valid.

Fermat’s Last Theorem was never the statement that “some future axioms of mathematics are inconsistent or the following equation is never satisfied by positive integers x, y, z, and n with n greater than 2.”

Fermat’s Last Theorem was simply the statement that, although there are equations like (3 squared) + (4 squared) = (5 squared), if you look at powers higher than 2, that does not happen. In other words, if x, y, z, and n are positive integers, and n is greater than 2, then (x to the power of n) + (y to the power of n) is not equal to (z to the power of n).

 

[PseuTonym]

According to the following video …

logic.harvard.edu/video.php?v=EFI_Magidor

… Wiles relied upon the assumption that there exist large cardinals. In other words, Wiles assumed not just ZFC set theory, but additional set theoretic assumptions.

However, you are unlikely to find any formulation of Fermat’s Last Theorem (the conjecture) that refers to axioms of set theory. After all, it is a conjecture of number theory, not set theory.

The video goes from 0 to 90.36, so if you don’t want to watch or listen to the whole video, then you need to know that you should watch or listen from 40.20 to 42.24

Here is a link to all of the talks in that workshop:
logic.harvard.edu/efi.php#multimedia

The project is called “Exploring the Frontiers of Incompleteness”, and is abbreviated EFI.

 

[Oreoracle]

You are free to define the word “conjecture” as you like, but you should be aware that it is a very non-standard definition.

Well, here’s what happens if you don’t accept that conjectures are made relative to a certain set of axioms.

Give me any conjecture you like. In addition to the axioms of ZFC set theory, I take the content of that conjecture as axiomatic. I “prove” the conjecture simply by noting that it is an axiom of the system.

No one does this because conjectures are made in the context of systems in which the axioms are much simpler than whatever is being conjectured. Take that context away and you could trivially prove anything.

For example, the twin primes conjecture in its current form does not specify what axioms may be used.

That is because, as you say, it is number theoretic. “Number theory” is more or less the set of results in one axiomatic system. Essentially the only differences between various axiomatizations of the integers are whether the Axiom of Choice is allowed, whether 0 is the least natural number or 1, and whether induction is a 2nd-order axiom or a 1st-order axiom schema. Besides a few details, everything else “goes without saying”. At this point, even the Axiom of Choice is probably assumed without saying by most of the mathematical community.

Wiles relied upon the assumption that there exist large cardinals. In other words, Wiles assumed not just ZFC set theory, but additional set theoretic assumptions.

However, you are unlikely to find any formulation of Fermat’s Last Theorem (the conjecture) that refers to axioms of set theory. After all, it is a conjecture of number theory, not set theory.

Okay, so Wiles has proven that Fermat’s Last Theorem holds within a particular axiomatic system, and not ZFC set theory. I don’t see how this damages my position. Indeed, if the axiomatic system is irrelevant to the conjecture, why specify at all which system one is using to prove it?

And it is helpful to have some context here. Fermat conjectured this result centuries ago, at a time in which most mathematicians held a Platonic view of mathematics. They did not see math as a human invention, but as something literally discovered in a manner analogous to an astronomer discovering a planet. Since they believed the objects of their study existed independently of axiomatic systems, it’s no wonder no one bothered to point out their axioms beyond saying that they were using the “correct” ones.

If, like them, you genuinely believe there are “correct” axioms, then tell me which is the “correct” geometry: Euclidean, elliptic, or hyperbolic? Are none of them correct?

 

[PseuTonym]

Mathematicians make assumptions like that all the time. For example, without knowing whether the twin primes conjecture is true or false, she can assume that it is true and show the consequences of that assumption, or she can assume that it is false and show the consequences of that assumption.

What do you mean by the words “without knowing whether the twin primes conjecture is true or false”? Are you attempting to see things from my point of view, or are you talking from your own point of view?

If you are attempting to see things from my point of view, then it is not helpful for you to demonstrate that the twin primes conjecture can be an assumption. I agree that it can be taken as an assumption in a given attempt to see what can be deduced. However I do not agree with your claim that an axiom is merely an assumption. That was your claim, not mine. Here is where you made that claim:

I don’t agree. First of all, people can always challenge an assumption. Secondly, if an axiom is not contradictory or if it does not contradict another axiom, it would not be false. It is just an assumption.

Now, reading that again, it sounds as though there is nothing to know. Applying the label “axiom” to the twin primes conjecture seems to magically transform it into something that cannot be false. That does not make any sense to me. I would appreciate it if you would explain your point of view. So far, I have seen little more than your desire to ignore everything in the first message of this thread except for the title. Geometry is considered to be part of mathematics, and the word “mathematics” is in the title, so you invoked it to get whatever conclusions you hoped to get about mathematics in general.

 

[Mount_Carmel]

I am a great fan of Maths, as I am of Science, and of Philosophy. BUT none of them, even Maths, can explain all things, locked as they are within the ‘physical and temporal’ dimensions of the universe. For me maths comes to a screaming halt [except in the case of the theoretical] when faced with the concept of infinity or eternity.

Although Georg Cantor did get as far as the idea of ‘infinite sets’, it did send him a bit barking mad - enough to spend some time in a psychiatric establishment.

IMHO, the examination of Islam is more a topic for Philosophical discourse.

ps. I highly rate John Lennox, who is a Prof. of Maths, amongst other things. -And whose YouTube ‘Lennox vs Hawking’ is a classic.