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The way that Mathematics and approaches this is, using the Axiomatic way. You can view this as a game: Let's make some assumtions / axioms, and see what w

The way that Mathematics and approaches this is, using the Axiomatic way. You can view this as a game: Let's make some assumtions / axioms, and see what we can build with it. The most popular axiom set is Zermelo-Fraenkel-style Set Theory [1], but there are other choices which have been explored. Note that, there a no claims about absolute truth or reality here. We just play a game in our minds. The most fundamental question about the game, that we can ask is "Is this game consistent". Goedel found out, somewhat unfortunately, we are not able to answer that question from within the game [2]. However, sofar no intrinsical inconsistencies (paradixies) have been found. Now, it turns out, that the ZF-Mathematics game is a very useful game, since it allows us to model certain aspects of reality. E.g. Newtons Mechanic, allows us to forecast how planets move in space. The Natural Sciences (Physics, Chem., Bio) and Engineering are essentially about how to model reality with mathematics. The way to do that is the empirical scientific method [3], where hypothesis are formed, experiments are designed and evidence is collected. So the question of "truth" does not really arise. It's just: (A) Mathematics is an interesting mind-game to play. (B) Mathematics has been used very successfully to model and predict our preceived reality. That's all you can ask for. [1] https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory [2] https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theorems https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theor... [3] https://en.wikipedia.org/wiki/Empirical_evidence https://en.wikipedia.org/wiki/Empirical_evidence

samirillian 7y ago
Yet the question of truth was indeed important to Godel, a lifelong Platonist. So it was the will to truth that brought us here in the first place Moreover, this tool-theory of truth that you proffer still runs on some concept of truthiness. All these system share a similar paradox to this trilemma: let's assume there is such a thing as truth, but the condition of that assumption seems to be the impossibility of proving it with any system produced by such a will to truth.
- heinrichhartman 7y ago
  It may well be that the search for "truth" has brought us here. However, we found out that "truth" is always a relative. It has to be rooted in assumptions/axioms. So I would say, it runs on, good choices of axioms and the ability to perform logical deductions. Set Theory is an attempt to come up with axioms that are so simple that no-one who is not completely out of it's mind would doubt their "truth". This is stuff like: - There exists a set - Two sets are equal if all their elements are equal - We can form subsets from sets, by picking out elements The only exception is the axiom of choice, which is less intuitively evident, but a technical necessity to build and reason about infinite sets like the real numbers.
btilly 7y ago
We just wind up with minor details such as almost all real numbers that exist cannot be in any way exactly described. So in what sense do they exist? (Proof: The reals are uncountable. The set of possible descriptions that might specify a real number is countable. Therefore almost all have no description.) But as long as we use words like "existence" in this weird way, everything works out.
- avz 7y ago
  That's OK. Assuming the universe is finite or countable, the number of all possible circumstances in which a real number needs to be described (e.g. a practical problem needing solution or somebody's curiosity needing satisfaction) is also countable. In other words, most real numbers will never come up in any situation in the entire history of a countable universe.
  - btilly 7y ago
    The same epistemological problem exists in a discrete context. My favorite example is https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theorem https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theo.... There is a proof by contradiction that any class of graphs with the property that it is closed under minors can be characterized by a finite list of forbidden minors. Which therefore means that there is a polynomial time algorithm for membership in the class. The classic example is that "planar graphs" have the necessary property, and there are exactly two forbidden minors. However it is a pure existence proof. And the following facts are also known. We do not have a general way to construct the forbidden minors. Or a way to find out how many there are. Or any way to verify whether any given list of them is complete. And not just, "We don't know how to do it." But, "No such algorithm exists." (More precisely that being able to answer those questions in general leads to contradictions.) So despite the entirely discrete nature of this problem (there are only a countable number of such classes), we find ourselves asserting the existence of a finite thing that we cannot find, bound the size of, or verify the correctness of if someone presented a reasonable candidate for our inspection. In what meaningful way does this thing exist?
zajio1am 7y ago
> The way that Mathematics and approaches this is, using the Axiomatic way. You can view this as a game: Let's make some assumtions / axioms, and see what we can build with it. It is often described in such way and you can see it like that in retrospect, but actually it is often intuitive grasp of mathematical structures first, axiomazation second. For example, natural numbers was understood (including number theory) for thousand years, but Peano axioms are from the end of 19. century. But that does not say there is 'absolute thruth', there are just infinite different graspable math structures and it is arbitrary (or pragmatic), which we consider interesting to develop.
- heinrichhartman 7y ago
  Sure, that's the historic route that we ended up at the place we are in. However, this is not the logical foundations that are used for current teaching and research. There has been a lot of effort, in particular in the late 19the century by folks like Hilbert, Goedel, Frege, etc. to put Mathematics on solid foundations. This is what we ended up with.
IngoBlechschmid 7y ago
More on this point of view ("formalism") and a provably unfixable issue (incompleteness) in these notes of mine: https://iblech.gitlab.io/bb/multiverse.html https://iblech.gitlab.io/bb/multiverse.html
- heinrichhartman 7y ago
  Just realized you are a student of Marc Nieper-Wißkirchen in Augsburg. Extend my greeting when you see him! He introduced me to Category Theory around 2006 in Mainz.
kazinator 7y ago
Before we can ask "is this consistent", we have to understand what consistency means. Basically, underneath axioms we have to have one more underpinning that is yet more immovable than axioms, and that is rationality. If someone has a problem with that, we tend not to engage them in debate. It may be that they thereby they remain unsatisfied, but that is their problem.
jmoss20 7y ago
I always read the second incompleteness theorem as important /because/ of the connection to truth. I think we (many, anyway) want to believe something like Leibniz' principle of sufficient reason: that every true statement has some sufficient reason for being so. We naturally interpret proofs in a formal system as that "sufficient reason". If we could prove from within a system that all and only consistent statements have a proof -- then that's great news! Maybe that system is Truth, or if it isn't maybe that's evidence some other system like it could be. So Godel's work crushed this hope. No system can prove itself consistent, so there is no system under which the principle of sufficient reason is true. Maybe then we shouldn't believe it! Then are there truths that are true for no reason at all...? But of course, there are many readings here -- many different roads to being thwarted by Godel :)

samirillian 7y ago

Yet the question of truth was indeed important to Godel, a lifelong Platonist. So it was the will to truth that brought us here in the first place Moreover, this tool-theory of truth that you proffer still runs on some concept of truthiness. All these system share a similar paradox to this trilemma: let's assume there is such a thing as truth, but the condition of that assumption seems to be the impossibility of proving it with any system produced by such a will to truth.

heinrichhartman 7y ago
It may well be that the search for "truth" has brought us here. However, we found out that "truth" is always a relative. It has to be rooted in assumptions/axioms. So I would say, it runs on, good choices of axioms and the ability to perform logical deductions. Set Theory is an attempt to come up with axioms that are so simple that no-one who is not completely out of it's mind would doubt their "truth". This is stuff like: - There exists a set - Two sets are equal if all their elements are equal - We can form subsets from sets, by picking out elements The only exception is the axiom of choice, which is less intuitively evident, but a technical necessity to build and reason about infinite sets like the real numbers.

btilly 7y ago

We just wind up with minor details such as almost all real numbers that exist cannot be in any way exactly described. So in what sense do they exist? (Proof: The reals are uncountable. The set of possible descriptions that might specify a real number is countable. Therefore almost all have no description.) But as long as we use words like "existence" in this weird way, everything works out.

avz 7y ago
That's OK. Assuming the universe is finite or countable, the number of all possible circumstances in which a real number needs to be described (e.g. a practical problem needing solution or somebody's curiosity needing satisfaction) is also countable. In other words, most real numbers will never come up in any situation in the entire history of a countable universe.
- btilly 7y ago
  The same epistemological problem exists in a discrete context. My favorite example is https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theorem https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theo.... There is a proof by contradiction that any class of graphs with the property that it is closed under minors can be characterized by a finite list of forbidden minors. Which therefore means that there is a polynomial time algorithm for membership in the class. The classic example is that "planar graphs" have the necessary property, and there are exactly two forbidden minors. However it is a pure existence proof. And the following facts are also known. We do not have a general way to construct the forbidden minors. Or a way to find out how many there are. Or any way to verify whether any given list of them is complete. And not just, "We don't know how to do it." But, "No such algorithm exists." (More precisely that being able to answer those questions in general leads to contradictions.) So despite the entirely discrete nature of this problem (there are only a countable number of such classes), we find ourselves asserting the existence of a finite thing that we cannot find, bound the size of, or verify the correctness of if someone presented a reasonable candidate for our inspection. In what meaningful way does this thing exist?

zajio1am 7y ago

> The way that Mathematics and approaches this is, using the Axiomatic way. You can view this as a game: Let's make some assumtions / axioms, and see what we can build with it. It is often described in such way and you can see it like that in retrospect, but actually it is often intuitive grasp of mathematical structures first, axiomazation second. For example, natural numbers was understood (including number theory) for thousand years, but Peano axioms are from the end of 19. century. But that does not say there is 'absolute thruth', there are just infinite different graspable math structures and it is arbitrary (or pragmatic), which we consider interesting to develop.

heinrichhartman 7y ago
Sure, that's the historic route that we ended up at the place we are in. However, this is not the logical foundations that are used for current teaching and research. There has been a lot of effort, in particular in the late 19the century by folks like Hilbert, Goedel, Frege, etc. to put Mathematics on solid foundations. This is what we ended up with.

IngoBlechschmid 7y ago

More on this point of view ("formalism") and a provably unfixable issue (incompleteness) in these notes of mine: https://iblech.gitlab.io/bb/multiverse.html https://iblech.gitlab.io/bb/multiverse.html

heinrichhartman 7y ago
Just realized you are a student of Marc Nieper-Wißkirchen in Augsburg. Extend my greeting when you see him! He introduced me to Category Theory around 2006 in Mainz.

kazinator 7y ago

Before we can ask "is this consistent", we have to understand what consistency means. Basically, underneath axioms we have to have one more underpinning that is yet more immovable than axioms, and that is rationality. If someone has a problem with that, we tend not to engage them in debate. It may be that they thereby they remain unsatisfied, but that is their problem.

jmoss20 7y ago

I always read the second incompleteness theorem as important /because/ of the connection to truth. I think we (many, anyway) want to believe something like Leibniz' principle of sufficient reason: that every true statement has some sufficient reason for being so. We naturally interpret proofs in a formal system as that "sufficient reason". If we could prove from within a system that all and only consistent statements have a proof -- then that's great news! Maybe that system is Truth, or if it isn't maybe that's evidence some other system like it could be. So Godel's work crushed this hope. No system can prove itself consistent, so there is no system under which the principle of sufficient reason is true. Maybe then we shouldn't believe it! Then are there truths that are true for no reason at all...? But of course, there are many readings here -- many different roads to being thwarted by Godel :)