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

[PseuTonym]

He does not care if those theorems will ever be useful for any purpose, he simply participates in a wonderful and complex mind game.

You could similarly say that an astronomer does not care whether or not knowing that the universe is expanding will ever be useful for any practical purpose outside of astronomy or cosmology. However, it would be strange to say that an astronomer “simply participates in a wonderful and complex mind game.” What the astronomer studies is real. So we again have the question of what you believe and what you are claiming about mathematics.

I should have explained more clearly why I compared a definition of medicine with your statement about what mathematics is. It is incorrect to identify mathematics with a tool, technique, construction, etc that is used in mathematics, just as it would be incorrect to define medicine as the art of listening to the beating of a heart with a stethoscope, while wearing a white coat. The problem here is not that we need to expand beyond a stethoscope and list a wide variety of instruments that doctors use. We should not list any instruments.

You could try to define astronomy as a subject area that uses telescopes to study the expanding universe. However, astronomy existed long before astronomers reached the conclusion that the universe is expanding. Astronomy existed before the telescope was invented.

 

[tomarin]

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.

Quite often the work of the most “pure” mathematicians, who scorn any real-world applications to their work and are in it only for the love of the subject, end up yielding unexpected boons to society such as computers. Math has a funny relationship with the “real world” in that way. I don’t know where Erdos stood in this debate over the objectivity of mathematical proofs, but I’m pretty sure he would scoff at your insistence that math is just a game with arbitrary rules.

 

[Pallas_Athene]

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?

The truth of the Goldbach conjecture is simply an open question. One cannot expect to restrict the method to prove a specific conjecture to a predefined branch of mathematics. In theory the best proof will enlighten us, why that theory is true, but that does not always happen.

Think of the 4-color theory, which has been proven by a “brute-force” method. Part of the proof was to show that a “sufficiently” large map can always be colored by 4 colors. After that the proof was finished by using computers, which verified that all “smaller” maps can actually be colored by 4 colors. This is not an elegant proof, and one can doubt if the programmers really examined all the possible scenarios.

Interestingly, the same problem on the surface of a torus (where 7 colors are needed) has been proven without resorting to computers.

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

I never heard of an effort to create a specific set of axioms, which are tailored to solve a specific problem. Actually it makes no sense to me. We already have a good set of axioms. The effort is to show that a certain theorem is the proper corollary of these axioms or not.

 

[Pallas_Athene]

But we can’t create a circle that has a ratio of circumference to diameter that isn’t pi, can we?

Sure we can. It is only true in the Euclidean geometry, but not in the Riemann (spherical) or the Bolyai-Gauss-Lobatchewski (hyperbolic) geometry.

Quite often the work of the most “pure” mathematicians, who scorn any real-world applications to their work and are in it only for the love of the subject, end up yielding unexpected boons to society such as computers. Math has a funny relationship with the “real world” in that way.

There are always people with funny views.

I don’t know where Erdos stood in this debate over the objectivity of mathematical proofs, but I’m pretty sure he would scoff at your insistence that math is just a game with arbitrary rules.

The rules of math are not really arbitrary. The natural numbers (positive integers) really come from “nature”, by observing that one apple and one other apple are two apples. Or one apple and one pear are two fruits. Numbers are the ultimate abstractions, when all the specifics of the objects are disregarded - with one exception - the number of them.

But we can extend the concept of numbers, and arrive at the set of “integers” - both positive and negative integers and the zero. There was a long and arduous process to convince people that “minus one” is a legitimate concept and that “zero” is also useful. There are all sorts of such extensions. The complex numbers, the vectors and matrices, etc… These extended “numbers” have very different characteristics. For example it is not necessarily true that a * b = b * a. And so on…

Math is a wonderful and fun game, and some of its results have excellent, practical applications.

 

[tomarin]

Math is a wonderful and fun game, and some of its results have excellent, practical applications.

I suppose I have no need then to learn about advanced math, or pay it any mind whatsoever.

 

[Pallas_Athene]

I suppose I have no need then to learn about advanced math, or pay it any mind whatsoever.

Of course. You can live your whole life without reading, too. Or listening to music, or whatever. Nevertheless, I will present you with a beautiful arrangement of the first 16 numbers.

  1    15    14      4
 12     6     7      9
  8    10    11      5
 13     3     2     16

There are so many interesting regularities that it is almost impossible to enumerate them. This “magic square” is about 3000 years old, it was created in India. Just the simplest, very basic property: in every row, every column and the two diagonals the sum of the numbers is 34. But this 34 pops up all over the place. Have fun to find a few more… if you are so inclined. If not, just forget about it.

 

[tomarin]

Of course. You can live your whole life without reading, too. Or listening to music, or whatever. Nevertheless, I will present you with a beautiful arrangement of the first 16 numbers.

  1    15    14      4
 12     6     7      9
  8    10    11      5
 13     3     2     16

There are so many interesting regularities that it is almost impossible to enumerate them. This “magic square” is about 3000 years old, it was created in India. Just the simplest, very basic property: in every row, every column and the two diagonals the sum of the numbers is 34. But this 34 pops up all over the place. Have fun to find a few more… if you are so inclined. If not, just forget about it.

Well thank you for that magic square. I happen to believe there is a lot of beauty in math, and that that beauty derives from something absolute, but I can see I lack the ability to convince you of that, so I don’t want to waste your or my own time in trying.

 

[Pallas_Athene]

Well thank you for that magic square. I happen to believe there is a lot of beauty in math, and that that beauty derives from something absolute, but I can see I lack the ability to convince you of that, so I don’t want to waste your or my own time in trying.

You are most welcome. Just a little interesting tidbit. For magic squares with the size of an odd number (3 x 3, 5 x 5, etc.) there is a very simple, universal method of creating them. For the ones, where the size is an even number (4 x 4, 6 x 6, etc…) there is no such universal method, but there are individual solutions for each even “n”.

 

[PseuTonym]

I never heard of an effort to create a specific set of axioms, which are tailored to solve a specific problem. Actually it makes no sense to me. We already have a good set of axioms.

I don’t know what you have in mind when you refer to a “specific set of axioms.” Nor do I know what you mean by “tailored to solve a specific problem.” Also, if you have never heard of it, then it is not clear to me why you mentioned it. Perhaps it would be helpful for you to quote something that you did not post in this thread, and describe your train of thought as you re-read it. You got something from something, but it is difficult for me to guess what you got, or where you got it from.

The effort is to show that a certain theorem is the proper corollary of these axioms or not.

What effort are you talking about? Are you saying that there is a goal in mathematics? Your use of terminology seems non-standard to me, so it would be helpful if you would provide some clarification. For example, did you mean “conjecture” when you wrote “theorem”? Ordinarily, a theorem has corollaries, and we do not speak of corollaries of an axiom.

However, more important than the terminology is what you mean by your final two words “or not.” Are you talking about a careful mathematical demonstration, using not merely ordinary step-by-step deductive reasoning, but also model theory, to show that it is impossible for a given conjecture to be deduced from a given list of axioms?

Or are you simply saying that one should not dogmatically assume that a given conjecture is true, and that one has the goal of either creating a proof that it is true or a proof that it is false? Of course, in some cases we distinguish between a “proof that the conjecture is false” and a “counter-example to the conjecture.” After all, a counter-example ordinarily looks quite different from a proof, even if in some sense a counter-example does simply prove something.

 

[Tomdstone]

Of course, in some cases we distinguish between a “proof that the conjecture is false” and a “counter-example to the conjecture.”

In either case, the conjecture is proven false.