The search for truth in mathematics

[PseuTonym]

Once upon a time, somebody had a flash of insight that provided knowledge of all truths about the positive integers. Many of those truths could be communicated to others as statements that were understood, but not necessarily accepted as true statements.

With help, the inspired person was able to discover a small list of statements (called “the Emperor’s Axioms”) that could be remembered, and that would be enough to deduce everything about positive integers that would be needed for practical purposes for thousands of years. The Emperor’s Axioms were taught to children throughout the world, and the world enjoyed consensus.

The Emperor’s Axioms were true, so they were quite different from the Emperor’s New Clothes. Occasionally, people imagined what would happen if somebody added anything new to the Emperor’s Axioms. They could foresee that consensus would be lost. A loss of consensus could mean only one thing: a future of continual disagreement about such a basic and fundamental thing as truths about positive integers.

Now, if disagreement about fundamentals provokes war, then the preservation of peace required that everybody continue to accept the Emperor’s Axioms as not merely true, but the whole truth, and the basis for any future deductions of truths of number theory. Philosophers invented and spread a philosophy intended to prevent people from trying to go beyond the Emperor’s Axioms.

According to that philosophy, mathematics is simply a formal game, so that anybody who seeks truth is engaged in self-deception, unless the search for truth occurs inside the box created by the Emperor’s Axioms. The Emperor’s Axioms were true by virtue of the meaning of the words they contained, and the meaning of those words was said to be completely determined by the Emperor’s Axioms.

One day, a man named Kurt Gödel proved that the The Emperor’s Axioms are incomplete. He believed that there are facts about intangible entities that could be discovered in future, and that we are not doomed to either fight endless war or remain trapped inside the box created by the Emperor’s Axioms. Was he right?

 

[Charlemagne_III]

Unlike Einstein, Kurt Gödel believed in a personal God. Have to give him credit for that.

But his remark about Islam is puzzling.

“I like Islam, it is a consistent [or consequential] idea of religion and open-minded.”

en.wikipedia.org/wiki/Kurt_G%C3%B6del#Religious_views

 

[Tomdstone]

Unlike Einstein, Kurt Gödel believed in a personal God. Have to give him credit for that.

But his remark about Islam is puzzling.

“I like Islam, it is a consistent [or consequential] idea of religion and open-minded.”

en.wikipedia.org/wiki/Kurt_G%C3%B6del#Religious_views

I guess a lot of people agree since Islam is the fastest growing religion in the world today.

 

[Tomdstone]

Once upon a time, somebody had a flash of insight that provided knowledge of all truths about the positive integers. Many of those truths could be communicated to others as statements that were understood, but not necessarily accepted as true statements.

With help, the inspired person was able to discover a small list of statements (called “the Emperor’s Axioms”) that could be remembered, and that would be enough to deduce everything about positive integers that would be needed for practical purposes for thousands of years. The Emperor’s Axioms were taught to children throughout the world, and the world enjoyed consensus.

The Emperor’s Axioms were true, so they were quite different from the Emperor’s New Clothes. Occasionally, people imagined what would happen if somebody added anything new to the Emperor’s Axioms. They could foresee that consensus would be lost. A loss of consensus could mean only one thing: a future of continual disagreement about such a basic and fundamental thing as truths about positive integers.

Now, if disagreement about fundamentals provokes war, then the preservation of peace required that everybody continue to accept the Emperor’s Axioms as not merely true, but the whole truth, and the basis for any future deductions of truths of number theory. Philosophers invented and spread a philosophy intended to prevent people from trying to go beyond the Emperor’s Axioms.

According to that philosophy, mathematics is simply a formal game, so that anybody who seeks truth is engaged in self-deception, unless the search for truth occurs inside the box created by the Emperor’s Axioms. The Emperor’s Axioms were true by virtue of the meaning of the words they contained, and the meaning of those words was said to be completely determined by the Emperor’s Axioms.

One day, a man named Kurt Gödel proved that the The Emperor’s Axioms are incomplete. He believed that there are facts about intangible entities that could be discovered in future, and that we are not doomed to either fight endless war or remain trapped inside the box created by the Emperor’s Axioms. Was he right?

Principia Mathematica was an attempt to describe a set of axioms and inference rules in symbolic logic from which you could deduce all mathematical truths. But i don’t see anything in Principia Mathematica which comes close to proving elementary results such as the Gauss Bonnet theorem.

 

[Rhubarb]

Principia Mathematica was an attempt to describe a set of axioms and inference rules in symbolic logic from which you could deduce all mathematical truths. But i don’t see anything in Principia Mathematica which comes close to proving elementary results such as the Gauss Bonnet theorem.

The Principia was Russell and Whitehead trying to show that the basics of arithmetic could be founded solely on logic without an appeal to something else. It’s mathematical logic, not pure math - a work of philosophy. It hoped to anewer an ancient problem about the nature of mathematical objects, like number, itself. Lots of giants had been working at this, Hilbert, Frege, Kant, JS Mill, Descartes, all the way back to Aristotle and Plato. The problem is that the math works - but trying to give an account as to why it works was controversial. I wish I had an answer - I suspect that the answer lies in the post-Logicism theories. And I suspect I spilled a lot of ink right now without saying much of anything.

 

[JuanFlorencio]

Once upon a time, somebody had a flash of insight that provided knowledge of all truths about the positive integers. Many of those truths could be communicated to others as statements that were understood, but not necessarily accepted as true statements.

With help, the inspired person was able to discover a small list of statements (called “the Emperor’s Axioms”) that could be remembered, and that would be enough to deduce everything about positive integers that would be needed for practical purposes for thousands of years. The Emperor’s Axioms were taught to children throughout the world, and the world enjoyed consensus.

The Emperor’s Axioms were true, so they were quite different from the Emperor’s New Clothes. Occasionally, people imagined what would happen if somebody added anything new to the Emperor’s Axioms. They could foresee that consensus would be lost. A loss of consensus could mean only one thing: a future of continual disagreement about such a basic and fundamental thing as truths about positive integers.

Now, if disagreement about fundamentals provokes war, then the preservation of peace required that everybody continue to accept the Emperor’s Axioms as not merely true, but the whole truth, and the basis for any future deductions of truths of number theory. Philosophers invented and spread a philosophy intended to prevent people from trying to go beyond the Emperor’s Axioms.

According to that philosophy, mathematics is simply a formal game, so that anybody who seeks truth is engaged in self-deception, unless the search for truth occurs inside the box created by the Emperor’s Axioms. The Emperor’s Axioms were true by virtue of the meaning of the words they contained, and the meaning of those words was said to be completely determined by the Emperor’s Axioms.

One day, a man named Kurt Gödel proved that the The Emperor’s Axioms are incomplete. He believed that there are facts about intangible entities that could be discovered in future, and that we are not doomed to either fight endless war or remain trapped inside the box created by the Emperor’s Axioms. Was he right?

Beautiful!

I have not had the chance to read and understand Gödel’s theorem. Was it based on the set of The Emperor’s Axioms?.. But anyway, I think that new axioms in a set will not affect the theorems that have already been deduced based on the previous limited set, because the new axioms will not enter as additional premises for the deduction of those theorems. If, for example, there are the axioms a1, a2 and a3, and we demonstrate theorem t1 based on a1 and a3, the theorem will not be affected if we introduce axiom a4 later, because it is not needed in the demonstration. So, I don’t foresee any disagreement here.

Another possibility besides adding new axioms to a given set is to substitute some of them for others, which is what happened in the case of the Euclidean and non-Euclidean geometries. Was this case covered in Gödel’s theorem? Because in this case I see possible disagreements coming, at least for a certain time.

Best regards
JuanFlorencio

 

[Bob_Crowley]

Gödel may be right in that the Emperor’s Axioms can’t be conclusively proved in all circumstances, but so far as I know, the Emperor’s Axioms work for all practical intents and purposes.

One and one will always equal two, and the addition, subtraction, multiplication and division tables seem to work with 100% accuracy.

 

[JuanFlorencio]

Gödel may be right in that the Emperor’s Axioms can’t be conclusively proved in all circumstances, but so far as I know, the Emperor’s Axioms work for all practical intents and purposes.

One and one will always equal two, and the addition, subtraction, multiplication and division tables seem to work with 100% accuracy.

Hi Bob!,

But axioms do not need any proof. That is why they are axioms!

Regards
JuanFlorencio

 

[Charlemagne_III]

I guess a lot of people agree since Islam is the fastest growing religion in the world today.

Indeed! The sword has a way of making fast converts.

The Nazi swords made a lot of converts too, and look how that ended.

 

[Tomdstone]

Indeed! The sword has a way of making fast converts.

The Nazi swords made a lot of converts too, and look how that ended.

Off topic but here’s an example of a Catholic priest who freely converted to Islam.
youtube.com/watch?v=nHBY2nGxJS8

 

[PseuTonym]

I have not had the chance to read and understand Gödel’s theorem. Was it based on the set of The Emperor’s Axioms?

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, Feferman 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.

We can make an analogy with Newtonian physics and simultaneity. What was Newton’s definition of simultaneity? I don’t think that he gave one. It is not as though Newton invented all concepts of physics. Einstein gave a definition of simultaneity precisely because he wanted to dig deeper into it and related concepts that are ordinary taken for granted. Applying this lesson to the story of The Emperor’s Axioms, we may consider the possibility that if the basis for Gödel’s incompleteness theorem is unreliable, then maybe there are some assumptions that are relied upon and not explicitly listed as axioms. However, if that is the situation, then I doubt that Gödel is to blame for it. As I said, as far as I know, he was working within standard meta-theory.

I think that new axioms in a set will not affect the theorems that have already been deduced based on the previous limited set, because the new axioms will not enter as additional premises for the deduction of those theorems. If, for example, there are the axioms a1, a2 and a3, and we demonstrate theorem t1 based on a1 and a3, the theorem will not be affected if we introduce axiom a4 later, because it is not needed in the demonstration. So, I don’t foresee any disagreement here.

There is a difference between what is needed from a theoretical point of view and what is used in practice. In education, there has to be some attention paid to what students can understand. It is possible to create proofs that students can check line-by-line, without understanding how the proofs were created, and without learning much from them. If there is an easy-to-understand way to prove some theorem, then students are usually not particularly interested in knowing alternative proofs that are much more difficult to understand and based on a more reliable basis because less is assumed.

Of course, more difficult alternative proofs are occasionally at least mentioned, and sometimes studied in detail. For example, in set theory, students will be told that some theorem can be proved without requiring the axiom of choice. In number theory, students may be told that a theorem has an “elementary” proof. An “elementary” proof is not a proof for students who are in elementary school and who have not yet reached high school. In this context, “elementary” means a proof that is based on something like the Peano postulates. In other words, there is a proof that assumes less than is usually assumed, but getting a strong result from weak assumptions ordinarily means overcoming significant challenges, not just for the person who originally invented the proof, but also for students who are trying to understand it.

When mathematicians invent powerful new methods for proving things more easily, they do not usually conduct opinion polls to determine whether or not students believe the assumptions that are used to prove that the methods work. What is invented today may be studied today by mathematicians, and in future by graduate students, and eventually perhaps by undergraduate students. The system is not really set up for skeptics.

Another possibility besides adding new axioms to a given set is to substitute some of them for others, which is what happened in the case of the Euclidean and non-Euclidean geometries. Was this case covered in Gödel’s theorem? Because in this case I see possible disagreements coming, at least for a certain time.

I don’t think that case was covered in Gödel’s theorem. My initial impression is that your question does not even make sense, because we are talking about the positive integers. However, I think that I understand your thought process, and I would like to provide an answer that actually explains something to you, rather than simply rejecting your question. I will need more time.

 

[JuanFlorencio]

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, Feferman 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.

Gödels theorem is in my queue, but I am patient to myself.

T
We can make an analogy with Newtonian physics and simultaneity. What was Newton’s definition of simultaneity? I don’t think that he gave one. It is not as though Newton invented all concepts of physics. Einstein gave a definition of simultaneity precisely because he wanted to dig deeper into it and related concepts that are ordinary taken for granted. Applying this lesson to the story of The Emperor’s Axioms, we may consider the possibility that if the basis for Gödel’s incompleteness theorem is unreliable, then maybe there are some assumptions that are relied upon and not explicitly listed as axioms. However, if that is the situation, then I doubt that Gödel is to blame for it. As I said, as far as I know, he was working within standard meta-theory.

Do you think that it is still possible to refute Gödel’s theorem?

T
There is a difference between what is needed from a theoretical point of view and what is used in practice. In education, there has to be some attention paid to what students can understand. It is possible to create proofs that students can check line-by-line, without understanding how the proofs were created, and without learning much from them. If there is an easy-to-understand way to prove some theorem, then students are usually not particularly interested in knowing alternative proofs that are much more difficult to understand and based on a more reliable basis because less is assumed.

Of course, more difficult alternative proofs are occasionally at least mentioned, and sometimes studied in detail. For example, in set theory, students will be told that some theorem can be proved without requiring the axiom of choice. In number theory, students may be told that a theorem has an “elementary” proof. An “elementary” proof is not a proof for students who are in elementary school and who have not yet reached high school. In this context, “elementary” means a proof that is based on something like the Peano postulates. In other words, there is a proof that assumes less than is usually assumed, but getting a strong result from weak assumptions ordinarily means overcoming significant challenges, not just for the person who originally invented the proof, but also for students who are trying to understand it.

When mathematicians invent powerful new methods for proving things more easily, they do not usually conduct opinion polls to determine whether or not students believe the assumptions that are used to prove that the methods work. What is invented today may be studied today by mathematicians, and in future by graduate students, and eventually perhaps by undergraduate students. The system is not really set up for skeptics.

Let me include in my response what you are saying: let’s suppose that we have the five axioms a1, a2, a3, a4, and a5. And we demonstrate theorem t1 in different ways: a) starting with axioms a1 and a2; b) starting with axioms a2, and a3; c) starting with axioms a1, a4 and a5. If later we add axiom a6 to our original set, theorem t1 will not be affected, because the new axiom is not required in any of the demonstrations. So, I foresee no disagreement.

T
I don’t think that case was covered in Gödel’s theorem. My initial impression is that your question does not even make sense, because we are talking about the positive integers. However, I think that I understand your thought process, and I would like to provide an answer that actually explains something to you, rather than simply rejecting your question. I will need more time.

That was just an example that came to my mind, Pseu Tonym! Can something like this happen in the theory of positive integers?

Good night!
JuanFlorencio

 

[tafan]

PseuTonym:

As far as I know, Gödel was working within standard meta-theory.

Godel was working with basic number theory (ie integers and the Plano axioms). This is as basic of a system as it comes and by showing it was incomplete, Godel was able to show more complex systems would also be incomplete.

JuanFlorencio:

Do you think that it is still possible to refute Gödel’s theorem?

Absolutely not.

 

[PseuTonym]

Let me include in my response what you are saying: let’s suppose that we have the five axioms a1, a2, a3, a4, and a5. And we demonstrate theorem t1 in different ways: a) starting with axioms a1 and a2;

If later we add axiom a6 to our original set, theorem t1 will not be affected, because the new axiom is not required in any of the demonstrations. So, I foresee no disagreement.

Near the very beginning of “The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory (1940)”, Godel proves a meta-theorem that he calls “M1. General existence theorem.” Often, in set theory there is an axiom schema of infinitely many axioms for the existence of classes, with one axiom for each formula P(x) that has x as a free variable. However, Godel is using a system of finitely many axioms.

Before Godel gets as far as stating M1, he writes the following:
"The following axiom (proved consistent by J. von Neumann (1929)) is not indispensable, but it simplifies considerably the later work:

Axiom D. If A is a non-empty class, then there exists a set u such that u is an element of A, and u and A are mutually exclusive in the sense (definition 1.23) that there does not exist a set t such that t is an element of A and t is an element of u."

If you want to see the version of things that dispenses with Axiom D and includes considerable complications as a result, then you will not find it in Godel’s paper. Your choices are to accept Axiom D or to look elsewhere.

Now, if you were looking in a textbook for students who are beginning to study set theory and who have little background in mathematics or deductive logic, then you might not see any reference to the option of dispensing with Axiom D. In other words, students might simply be presented with Axiom D as something that they are required to accept. If, for whatever reason, a student believes that Axiom D is implausible, then the student might eventually lose the motivation to study proofs that rely on the assumption that Axiom D is true.

The important point is that students might not suspect that there was a historical event of your axiom a6 or Godel’s axiom D being added to a pre-existing system of axioms. Students might have no way of knowing that axiom a6 or axiom D is not required, because they might never see any reference to the existence of alternative, more complicated demonstrations that dispense with axiom a6 or axiom D.

The danger is that mathematicians might eventually accept some axiom that is actually false, and people who can intuitively sense something fishy about the axiom would not be able to challenge the status quo. True believers would be more successful in mathematics, and they would be doing the research, writing the textbooks, and teaching the classes. Eventually the false axiom might be deeply embedded in things, and reconstructing mathematics to avoid it could require an enormous amount of work plus new ideas, and the new ideas might not arrive on schedule when they are wanted.

 

[Tomdstone]

The danger is that mathematicians might eventually accept some axiom that is actually false, and people who can intuitively sense something fishy about the axiom would not be able to challenge the status quo. .

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.

 

[Oreoracle]

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

You are worried that adding axioms would cause disagreement over the set of integers. However, the integers are defined as members of a set that satisfies a list of axioms. In other words, if you change the axioms, you change what “integer” means.

So if I claim the integers have one property and you disagree because our axioms are different, there is no actual “disagreement”. 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.

 

[PseuTonym]

However, the integers are defined as members of a set that satisfies a list of axioms. In other words, if you change the axioms, you change what “integer” means.

Philosophers in the story make that claim:

Philosophers invented and spread a philosophy intended to prevent people from trying to go beyond the Emperor’s Axioms.

According to that philosophy, mathematics is simply a formal game, so that anybody who seeks truth is engaged in self-deception, unless the search for truth occurs inside the box created by the Emperor’s Axioms. The Emperor’s Axioms were true by virtue of the meaning of the words they contained, and the meaning of those words was said to be completely determined by the Emperor’s Axioms.

 

[PseuTonym]

First of all, people can always challenge an assumption.

What is your claim? Are you talking about the law in some jurisdiction? Are you talking about the practical consequences of attempting to challenge an assumption? I would say that if the practical effect of a person attempting to challenge an assumption is that authorities label the person as stupid or insane, and no details can be later discovered about what the person who attempted to challenge the assumption had in mind, then the attempt failed. The assumption was not actually challenged if nobody who is interested in the question can discover anything about the train of thought that gave the person reason to suspect that the assumption might be false.

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.

In practice, we do not know whether or not any given non-trivial list of axioms is consistent.

However, let us assume for the sake of argument that some particular conjecture such as the twin primes conjecture can be neither proved nor disproved from some particular formalization of generally accepted mathematics. In that case, would you say that we have the following two options? The first option is to freely and arbitrarily assert that the twin primes conjecture is true. The second option is to freely and arbitrarily assert that the twin primes conjecture is false.

It is just an assumption.

You are omitting something: what it is not. If I say that a diamond is just an expensive part of jewelry, my statement is in a sense inaccurate. Diamond is an extremely hard material. It has special properties not shared with other materials, and certainly that hardness is not shared with gold, another popular material for jewelry. When we use the word “just” in a sentence such as “a diamond is just an expensive part of jewelry”, we are denying something about diamonds, but there is no way to even guess what is denied unless there is more contextual information or more information provided by the person who is making the claim that it is “just” that (and nothing else).

 

[Tomdstone]

I would say that if the practical effect of a person attempting to challenge an assumption is that authorities label the person as stupid or insane, and no details can be later discovered about what the person who attempted to challenge the assumption had in mind, then the attempt failed.

I don’t know of any “authorities” who would label a person insane if she challenged the Euclidean parallel postulate.

 

[Tomdstone]

However, let us assume for the sake of argument that some particular conjecture such as the twin primes conjecture can be neither proved nor disproved from some particular formalization of generally accepted mathematics. In that case, would you say that we have the following two options? The first option is to freely and arbitrarily assert that the twin primes conjecture is true. The second option is to freely and arbitrarily assert that the twin primes conjecture is false.

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.