The search for truth in two branches of mathematics: number theory and set theory

[PseuTonym]

I have nothing to say, in this first message of this thread, about the search for truth in set theory. However, please do not let that prevent you from replying to this thread with your own comments on the search for truth in set theory. So, let us begin, knowing that, in this first message of the thread, the topic is the search for truth in number theory. 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, it was discovered that the The Emperor’s Axioms are incomplete. This discovery suggests that there may be 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.

 

[Tomdstone]

I have heard of the Peano axioms, but not the Emperor’s axioms.

 

[Pallas_Athene]

According to that philosophy, mathematics is simply a formal game…

Well, mathematics IS a formal game, one of innumerable other ones. The concept of “truth” in any formal system simply means that a proposition can be derived from the accepted axioms, by using the “grammar” of transformation rules. There is no need for it to have a referent in the “outside” world.

Any game with any set of artificial rules (axioms) and artificial grammar is an axiomatic system, with its “true” propositions. Take chess, for example. The “axioms” are the board and the pieces on their starting positions. The “grammar” is rules how the pieces can be moved. A proposition is set of pieces in some specific arrangement. This proposition is “true” if there can be a set of moves which will lead to the arrangement of the pieces starting from the original setup.

But the board does not have to be the size of 8 by 8. The set of pieces can be extended, with new rules of movement. Such axioms create fairy-chess games. The question of “which is true”, the original or the fairy-chess game is not a valid question.

Any set of axioms can be used with any grammar rules - provided that there is internally inconsistent rules.

Going back to mathematics. If we define an extended set of “integers” as (a + b * z) where “z” is the “square root of minus five” and “a” and “b” are regular integers, it will be a superset of the “normal” arithmetics - when “b” is set to zero. In the “normal” set of integers there is a theorem, which states that every positive integer can be factored into a unique set of prime numbers. In the extended set this theorem is no longer true. The number 6 can be factored as 2 * 3, or (1+z)*(1-z), which are clearly different. Now one cannot ask if the theorem of the unique factorization is “true” or not. It all depends on the axioms selected.

I have no idea what your original intent was in your OP, but I thought I can introduce a comment.

 

[Tomdstone]

Well, mathematics IS a formal game, one of innumerable other ones. The concept of “truth” in any formal system simply means that a proposition can be derived from the accepted axioms, by using the “grammar” of transformation rules. There is no need for it to have a referent in the “outside” world.

Any game with any set of artificial rules (axioms) and artificial grammar is an axiomatic system, with its “true” propositions. Take chess, for example. The “axioms” are the board and the pieces on their starting positions. The “grammar” is rules how the pieces can be moved. A proposition is set of pieces in some specific arrangement. This proposition is “true” if there can be a set of moves which will lead to the arrangement of the pieces starting from the original setup.

But the board does not have to be the size of 8 by 8. The set of pieces can be extended, with new rules of movement. Such axioms create fairy-chess games. The question of “which is true”, the original or the fairy-chess game is not a valid question.

Any set of axioms can be used with any grammar rules - provided that there is internally inconsistent rules.

Going back to mathematics. If we define an extended set of “integers” as (a + b * z) where “z” is the “square root of minus five” and “a” and “b” are regular integers, it will be a superset of the “normal” arithmetics - when “b” is set to zero. In the “normal” set of integers there is a theorem, which states that every positive integer can be factored into a unique set of prime numbers. In the extended set this theorem is no longer true. The number 6 can be factored as 2 * 3, or (1+z)*(1-z), which are clearly different. Now one cannot ask if the theorem of the unique factorization is “true” or not. It all depends on the axioms selected.

I have no idea what your original intent was in your OP, but I thought I can introduce a comment.

Are z, 1+z, 1-z prime numbers?

 

[Pallas_Athene]

Is z a prime number?
Are z, 1+z, 1-z prime numbers?

In that system they are, since they have no non-trivial divisors. The definition of a prime number is that it only has two trivial divisors, namely “1” and “itself”. z = 0 + 1 * z, which means that it is of the form “a + b * z” which is axiomatically defined to be the notion of “integers”.

This “strange” type of integers is just a subset of the complex numbers, which are defined as “a + b * i”, where “i” is the square root of minus one.

 

[Tomdstone]

In that system it is, since it has no non-trivial divisors. The definition of a prime number is that it only has two trivial divisors, namely “1” and “itself”. z = 0 + 1 * z, which means that it is of the form “a + b * z” which is axiomatically defined to be the notion of “integers”.

No, it is not. That is not right according to the standard definition. You have redefined what you mean by a prime number, so of course the standard prime number theorem will have to be reevaluated in light of your new definition. In the standard definition, a prime number is from the set of natural numbers; i.e., from the set {1,2,3,4,5,6,7,8,9,…}.
It makes no sense to say that the prime number theorem is not true in the extended set, because you have redefined what is meant by a prime number.

 

[Pallas_Athene]

No, it is not. That is not right according to the standard definition. You have redefined what you mean by a prime number, so of course the standard prime number theorem will have to be reevaluated in light of your new definition. In the standard definition, a prime number is from the set of natural numbers; i.e., from the set {1,2,3,4,5,6,7,8,9,…}.
It makes no sense to say that the prime number theorem is not true in the extended set, because you have redefined what is meant by a prime number.

We extend the concept of integers. That is just another extension, a new set of axioms. You can extend the definition of numbers to incorporate vectors like [a[sub]1[/sub], a[sub]2[/sub], a[sub]3[/sub],…, a[sub]n[/sub]]. In this system division is not even possible to define. The point is that in mathematics everything depends on the axioms we happen to define. Mathematics is the pen-ultimate relativistic system. Are you familiar with how a theorem is stated? “Let a, b, whatever, is supposed to be true. Then x, y, z, etc… will follow”. It is always of the form “IF X, THEN Y”. There are no “absolute” truths in mathematics. Everything is ultimately contingent upon the axioms. And we can create new sets of axioms at our leisure. They may not be useful for anything, but that is not relevant. Mathematics is just a very complex, elaborate game. Fun to do it, for sure.

 

[Tomdstone]

We extend the concept of integers…

You can’t expect a theorem which holds in the standard definition to hold in the extended case.

 

[Tomdstone]

We extend the concept of integers…

You are redefining things, so you can’t expect a theorem which holds in the usual situation to hold in the extended case.

 

[inocente]

I have nothing to say, in this first message of this thread, about the search for truth in set theory. However, please do not let that prevent you from replying to this thread with your own comments on the search for truth in set theory. So, let us begin, knowing that, in this first message of the thread, the topic is the search for truth in number theory.

And after that your OP seems, word for word, exactly the same as the OP of a thread you started in February, “The search for truth in mathematics”.

What are you hoping for that you didn’t get from the respondents to your previous thread?

 

[Pallas_Athene]

You are redefining things, so you can’t expect a theorem which holds in the usual situation to hold in the extended case.

It all depends on the specifics of the new axiomatic system. Sometimes the theorems are true in both, sometimes they are not. The point is that there are no “absolute” mathematical truths, they are all the corollaries of axioms.

 

[Tomdstone]

The point is that there are no “absolute” mathematical truths, they are all the corollaries of axioms.

I disagree since I believe that there are absolute mathematical truths which are true given any working non-trivial axiomatic system. I think it is absolutely and always true that: 1 does not equal 0. If you deny this then kindly give us a non-trivial, useful, working mathematical system where: 1=0.

 

[Pallas_Athene]

I disagree since I believe that there are absolute mathematical truths which are true given any working non-trivial axiomatic system. I think it is absolutely and always true that: 1 does not equal 0. If you deny this then kindly give us a non-trivial, useful, working mathematical system where: 1=0.

If an axiomatic system would contain a logical contradiction, it would be invalid. Of course the trivial proposition of 1 = 1 would be true in any axiomatic system, but that is just a different format for the trivial “law of identity”, which is one of the three axioms of logic.

What is the “truth” value of 1 + 1 = 2 and 1 + 1 = 1? Guess what, both are correct. The first one uses the regular algebra, the second one uses the Boolean arithmetic.

 

[PseuTonym]

I have no idea what your original intent was in your OP, but I thought I can introduce a comment.

Your comments are perfectly on topic. Thank you for replying. Your reply was quite elaborate and I am concerned that you might consider it dismissive that I reply to only some parts of what you wrote. However, please take these words as a kind of first try. I will respond more thoroughly later if you wish.

I respectfully disagree with the opinion that you have expressed in your comments, and my main hope for this thread is that both of us make our discussion or debate as clear as possible. Please ask me to clarify anything that you consider vague, ambiguous, or otherwise unclear.

Going back to mathematics. If we define an extended set of “integers” as (a + b * z) where “z” is the “square root of minus five” and “a” and “b” are regular integers, it will be a superset of the “normal” arithmetics - when “b” is set to zero. In the “normal” set of integers there is a theorem, which states that every positive integer can be factored into a unique set of prime numbers. In the extended set this theorem is no longer true. The number 6 can be factored as 2 * 3, or (1+z)*(1-z), which are clearly different. Now one cannot ask if the theorem of the unique factorization is “true” or not. It all depends on the axioms selected.

Yes, that is a popular example of how unique factorization is, in some rings, not satisfied. I believe that an early attempt to prove Fermat’s Last Theorem failed precisely because a mathematician was using a ring and presumed but neglected to check that the ring satisfied unique factorization.

Now, if mathematics is simply the study of formal systems, then we might wonder how it is possible to use some ring such as the Gaussian integers (which does have applications in number theory) to prove theorems about the ordinary (or as you call them “regular”) integers. The answer seems to be that mathematicians use the full resources of mathematics to prove theorems in number theory. However, if number theory were a formal system, then mathematicians would be able to use only the axioms of that formal system to prove theorems in number theory.

I do not want to get into a debate about the boundaries between different branches of mathematics because it could easily degenerate into a mere disagreement over labels or words. However, it seems necessary for me to at least mention for the benefit of everybody reading this that your counter-example to unique factorization that uses z = “square root of minus five” and (2 times 3) = (1+z)*(1-z) is in the realm of abstract algebra rather than number theory. Because it is a counter-example rather than a theorem, it is doubtful that it has any application to number theory. Number theory is not the study of anything and everything that gets the label “number”, but is specifically about the non-negative integers.

Well, mathematics IS a formal game, one of innumerable other ones. The concept of “truth” in any formal system simply means that a proposition can be derived from the accepted axioms, by using the “grammar” of transformation rules. There is no need for it to have a referent in the “outside” world.

It is not clear what you are assuming and what you are trying to demonstrate. What you say about formal systems might be perfectly accurate, but how does it allow us to reach any conclusion about number theory or set theory? It remains for you to demonstrate a connection between mathematics (specifically number theory or set theory) and formal systems. After all, it would be a strange definition of the word “mathematics” that placed number theory altogether outside the realm of mathematics.

 

[PseuTonym]

What are you hoping for that you didn’t get from the respondents to your previous thread?

Studying the previous thread, I see a denial that there is a truth to be searched for. I do not see any point of view that goes beyond that mere denial. I see tactics and strategies, with heavy use of control of the focus of attention. For example, previously there was a heavy focus on geometry, apparently licensed by the title of the previous thread. I do not see how I could have anticipated that problem, but that experience indicated that it was necessary to revise the title of the thread.

I am hoping to see a clear point of view expressed, a point of view that goes beyond the plain, unexplained denial that there could be such a thing as true foundations for number theory or set theory that one might search for.

 

[PseuTonym]

Well, mathematics IS a formal game, one of innumerable other ones. The concept of “truth” in any formal system simply means that a proposition can be derived from the accepted axioms, by using the “grammar” of transformation rules. There is no need for it to have a referent in the “outside” world.

Any game with any set of artificial rules (axioms) and artificial grammar is an axiomatic system, with its “true” propositions. Take chess, for example. The “axioms” are the board and the pieces on their starting positions. The “grammar” is rules how the pieces can be moved. A proposition is set of pieces in some specific arrangement. This proposition is “true” if there can be a set of moves which will lead to the arrangement of the pieces starting from the original setup.

That is an admirably brief and clear description of what a formal system is. However, formal systems belong in a family that includes techniques, methods, apparatus, constructions, etc.

Consider an analogy: instead of trying to specify what mathematics is, we can try to say what medicine is: “the science and art dealing with the maintenance of health and the prevention, alleviation, or cure of disease.” That is not a perfect definition, but it has the merit of focusing on goals rather than methods.

If we focus on methods, then we will discover that the presentation depends upon the historical era, and that in order to maintain complete accuracy in our description, we require increasing complication and complexity, introducing many distinctions, and basically teaching either the history of medicine or some aspect of contemporary medical principles and practice. For a long time, any description of medical practice would have included bloodletting as an important part of medical practice. It is not a significant part of medical practice today.

If we look back at the chunk of text that you wrote and I quoted, then we observe that you did not actually say what the goal of mathematics is. Instead, you attempted to define mathematics in terms of some technique, method, apparatus, or construction.

But the board does not have to be the size of 8 by 8. The set of pieces can be extended, with new rules of movement. Such axioms create fairy-chess games. The question of “which is true”, the original or the fairy-chess game is not a valid question.

I agree with you: it makes no sense to ask whether or not regular chess or fair-chess is true. I believe that the explanation of why it makes no sense is that each kind of chess is basically an example of a construction, rather than a statement.

So let us consider a statement.

Specifically, consider the statement that is known as the “twin primes conjecture” (tpc): for every positive integer n, there exists a number p such that p is greater than n, and both p and p+2 are prime. The tpc is either true or false. It is conceivable that the generally accepted foundations of mathematics do not supply enough information to determine whether or not the tpc is true. However, in that situation it would not be justifiable to reach the conclusion that “is the tpc true?” is not a valid question.

The history of mathematics does not begin with the logical foundations. The future of mathematics is not restricted to using the techniques – and accepting the beliefs – of early 21st century mathematics.

Today it is believed that Fermat’s Last Theorem is true. However, if a counter-example were discovered, then it would be possible for people around the world to confirm that the counter-example is genuine. Once the values x, y, z, and n for the counter-example were known, it would be a simple matter of computation, with no possibility for controversy. In contrast, the proof that Andrew Wiles created is very complicated and depends upon believing that various assumptions are actually true.

I deliberately selected the title of this thread to exclude geometry as a topic for discussion in this thread. However, if we are to discuss the history of mathematics, then we cannot ignore Euclid’s Elements. The first point is that geometry existed before Euclid’s Elements. It would be absurd to say that if Euclid had omitted all of the proofs, then he would have contributed nothing to mathematics. The second point is that Pasch’s Postulate was discovered by Moritz Pasch in the late 1800s, but it was implicitly relied upon in Euclid’s Elements.

Here is a rather slow-moving video about Pasch’s Postulate:
youtube.com/watch?v=YlbvNIuzy8Y&list=PLDAPfGC8G5lchl0yjjio8Qd_z196dLqJq&index=20

 

[inocente]

Studying the previous thread, I see a denial that there is a truth to be searched for. I do not see any point of view that goes beyond that mere denial. I see tactics and strategies, with heavy use of control of the focus of attention. For example, previously there was a heavy focus on geometry, apparently licensed by the title of the previous thread. I do not see how I could have anticipated that problem, but that experience indicated that it was necessary to revise the title of the thread.

I am hoping to see a clear point of view expressed, a point of view that goes beyond the plain, unexplained denial that there could be such a thing as true foundations for number theory or set theory that one might search for.

All I knew about foundations of math was that Bertrand Russell & co. did some work. Looking that up, I was led to an article on David Hilbert which says:

Although it is not possible to formalize all mathematics, it is possible to formalize essentially all the mathematics that anyone uses. In particular Zermelo–Fraenkel set theory, combined with first-order logic, gives a satisfactory and generally accepted formalism for essentially all current mathematics.

I’m not a mathematician and that put me straight out of my depth, so over to you guys.

en.wikipedia.org/wiki/Hilbert’s_program#Hilbert.27s_program_after_G.C3.B6del
plato.stanford.edu/entries/set-theory/#AxiZFC

 

[Pallas_Athene]

If we look back at the chunk of text that you wrote and I quoted, then we observe that you did not actually say what the goal of mathematics is. Instead, you attempted to define mathematics in terms of some technique, method, apparatus, or construction.

Let me share with you what my favorite science fiction writer / philosopher Stanislaw Lem wrote in his wonderful book: “Summa Technologiae” (the chosen title is not a coincidence ). en.wikipedia.org/wiki/Summa_Technologiae

He writes about a “crazy” tailor, who sits in his workshop, and dreams up all sorts of creatures. He then makes clothes for them, and stores the clothes in his huge warehouse. When a customer comes in and asks for an outfit for his “pet”, the tailor usually can supply the necessary clothes. Sometimes, he does not have a prepared outfit, so he asks the customer for the specification of his pet, and attempts to create the goods. Usually he succeeds, but not always.

This tailor is the mathematician. He dreams up all sorts of formal systems (theorems). He does not care if those theorems will ever be useful for any purpose, he simply participates in a wonderful and complex mind game.

Just an example. Linear algebra (vectors and matrices) was “dreamed up”. It was a nice but empty construct, totally useless, until some people whose “pet” was quantum mechanics knocked on the door, and asked for a set of clothes (methods). It just so happened that linear algebra was fitting and became useful. Later on linear (and non-linear) programming (operations research) came onto the scene, and it became a new “customer” for the same “clothes”.

For the time being we can see no practical use for number theory. Whether the twin-prime conjecture is true or not has no significance in “real life”. It is just a great challenge for the people. If and when the problem of Riemann’s zeta function will be solved, that would also prove the twin-prime problem.

An anecdote for you. There was a great Hungarian mathematician, Paul Erdős. He was an atheist, but he firmly believed in the existence of “The Book”, which contains the most elegant, most beautiful proofs of all the theorems. When he encountered a particularly elegant proof, he always said: “this is straight from the Book”, and there was no greater praise from him.

 

[PseuTonym]

Whether the twin-prime conjecture is true or not has no significance in “real life”. It is just a great challenge for the people.

Are you acknowledging that there is a fact of the matter as to whether or not the twin-primes conjecture is true, even if the generally accepted foundations of mathematics are not sufficient to support either the claim that the tpc is true or the claim that the tpc is false?

Previously, it seemed that the view you were expressing allowed for the possibility that the twin-primes conjecture does not have a definite truth value. Previously it seemed that you were expressing the view that a given conjecture in mathematics has a truth value relative to whatever axioms one might choose, and that there is no restriction on what axioms are chosen, except that the system of axioms needs to be logically consistent, an internal criterion. For example, look at the following:

Well, mathematics IS a formal game, one of innumerable other ones. The concept of “truth” in any formal system simply means that a proposition can be derived from the accepted axioms, by using the “grammar” of transformation rules. There is no need for it to have a referent in the “outside” world.

The question of “which is true”, the original or the fairy-chess game is not a valid question.

Now one cannot ask if the theorem of the unique factorization is “true” or not. It all depends on the axioms selected.

Do you acknowledge that among the goals of mathematics is to discover true axioms for number theory and set theory?

On the topic of practical applications of mathematics, I agree that it is possible to do legitimate mathematical research that does not yet have any known practical applications.

He dreams up all sorts of formal systems (theorems). He does not care if those theorems will ever be useful for any purpose, he simply participates in a wonderful and complex mind game.

For the time being we can see no practical use for number theory.

However, the topic of practical applications is quite different from the topic of searching for truth. Some people will refrain from searching for truth if they foresee no practical benefit. However, others will persist.

The most effective way to prevent people from searching for truth is to persuade them that there is no truth to be searched for. Of course, if you sincerely believe that there is no truth to be searched for, then your motive might simply be to convert them to your beliefs. However, we might nevertheless wonder who or what influenced you to adopt your belief. If it is part of the Zeitgeist, then how and why did it become part of the Zeitgeist?

 

[tomarin]

We extend the concept of integers. That is just another extension, a new set of axioms. You can extend the definition of numbers to incorporate vectors like [a[sub]1[/sub], a[sub]2[/sub], a[sub]3[/sub],…, a[sub]n[/sub]]. In this system division is not even possible to define. The point is that in mathematics everything depends on the axioms we happen to define. Mathematics is the pen-ultimate relativistic system. Are you familiar with how a theorem is stated? “Let a, b, whatever, is supposed to be true. Then x, y, z, etc… will follow”. It is always of the form “IF X, THEN Y”. There are no “absolute” truths in mathematics. Everything is ultimately contingent upon the axioms. And we can create new sets of axioms at our leisure. They may not be useful for anything, but that is not relevant. Mathematics is just a very complex, elaborate game. Fun to do it, for sure.

But we can’t create a circle that has a ratio of circumference to diameter that isn’t pi, can we? Or are you saying a circle is a culturally-constructed concept? If so I have to disagree.