Mephist

Well, I believe that some form of artificial intelligence capable of recognizing interesting theorems should be possible one day. But the interesting ...

Maybe I didn't understand what you meant. For what I know of artificial intelligence, the main point is pattern recognition. But you spoke about "cont...

I wouldn't call that kind of logical deductions "intelligence". This is the kind of deductions that a computer can do automatically. Logic is basicall...

If you can set up a proposition that refuse analysis, it means that your logic is wrong. The purpose of logic (the reason it was created) is to rule o...

OK, I am not an expert either :-) For what I can say for sure, there is a full formalization of real numbers in COQ standard library (https://coq.inri...

Wait a moment: I am advocating Martin-Löf type theory, and one of it's particular versions called "Calculus of Inductive Constructions" that is, as th...

I am glad that you replied to my post, so I can explain in more detail my point of view :smile: I don't believe that physics is founded on ZF Set theo...

OK, your are right. Let's stick to a formal logic that uses only the forall-exists language. But we have to keep in mind that the meaning of "forall" ...

An idea is arbitrary data, such as a picture. For this you need a memory to store information, because there are a lot of possible combinations of sha...

HOTT is Martin-Lof Type Theory with higher inductive types, but with the addition of Voevodsky's "univalence axiom". So, it's not so simple: first of ...

First of all, let's look for a definition of "constructive mathematics", because I have the impression that we are not speaking about the same thing. ...

Of course I have! I was only waiting for somebody to ask me to announce the truth to the world :smile: :joke: The answer of course is based on modern ...

Yes, what I meant is that some parts of mathematics are "interesting" and some are not. And I think this distinction can be made internally to mathema...

In my opinion we don't call mathematics what is patternless. It's not clear for me what a pattern is, but I would argue that nobody would consider a m...

Well, my idea is that there is a "map" that exists in some kind of Platonic world that is taken as a model by nature. Parts of this map are taken as a...

You are right. I think I should be more careful with this kind of statements. I wrote this without really checking the recursive structure generated b...

I believe it's not only from observations of nature that mathematics takes inspiration. An obvious example: Mandelbrot set is not present in nature, b...

OK, probably this is not the conventional point of view in mathematics, but I'll try to explain my point of view: In ZF Set theory you have the "axiom...

OK, I have no convincing argument against this point of view, except that if irrational numbers do not exist a great part of mathematics becomes more ...

All irrational numbers are defined as infinite objects (usually infinite sums or products): pi = lim ( sum for k=1..n ((-1)^(k+1) / (2*k - 1)) ) n->in...

I understand you point of view: I think this was the common point of view of mathematicians from ancient Greeks until the beginning of XVII century: y...

Well, not exactly... ? - ? = 0 is not true, because ? - ? is not a term of the language. Let me explain better: let's take the function "int", defined...

First of all, I used the term "finitary model" meaning "a model that is built from a finitary theory", but probably it's a misused term. So, let's spe...

If you want to use formal logic (that, for what I know, is the only form of logic that we can trust for sure), you have to define the term "objective ...

Yes, exactly. The world and our mind have forms that are similar to interesting mathematical objects because these forms are in some way special, and ...

This is a post from 5 days ago, but it's an interesting subject so I'll reply to it now. I think that we could find the reason to favor numbers and ge...

I read the book that you suggested me (thanks for the link!) I think that the "trick" that avoids inconsistency is that the inverse of infinite is not...

I'll give you an example: the game of chess exists "ouside of our minds" as list of possible "positions" and a list of allowed "moves" that the player...

Yes, but even if the algorithms were subject to any arbitrary changes by the author of the game, he would still be able to prove the same theorems of ...

Let's try to make more precise the distinction between "mathematical" and "physical" law: suppose that our alien was not a real alien, but a character...

Sorry, I don't know transfinite aritmetics.. :sad: But if you have some good links to documents that explain what is it I would be interested!

Well.. I don't like it too, but nobody has shown that is inconsistent yet, and it's used since a very long time. So, I would guess that it's not incon...

OK, I didn't understand at first that you wanted to use it as a "fasle proof". I agree, this proof is surely not acceptable for a number or reasons. F...

The conclusion could even be that "the measure of the set of all sets does not exist" this is an assumption too.

True. But you can't use such number system (integers plus "infinite") to define a "size" of sets that respects the properties 1 to 4.

Well, the problem is to give a definition of "size" for sets that you cannot count. And we want this definition to have some "reasonable" properties: ...

Sorry, I wanted to write "finitary", in the sense of "recursively enumerable" (of course not finite, if you can build natural numbers with sets) Howev...

Yes, exactly. not in "Principia Mathematica" ( my mistake ). OK, I am glad to hear that you agree :smile: I'll add even something else, that probably ...

What I wanted to say is that Russel's paradox invalidates the use of "naive set theory", that is the kind of set theory used on Principia Mathematica

OK, maybe I wanted to make it too simple :smile: ZF Set theory is another way to limit the use of "naive set theory", in my opinion much more complica...

From (https://en.wikipedia.org/wiki/Russell%27s_paradox): "In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zerm...

Cantor's "diagonal" theorem on the existence of an infinite hierarchy of infinities can be expressed in a quite convincing way: "for every set A, the ...

Well, that's not exactly what I had in mind saying that "the fact that two independent civilizations "invent" the same mathematical theorem is a proof...

Well, what I meant to ask was is if could exist (or if somebody invented) any "automatic" and "objective" way to recognize meaningful mathematical the...

I think that we could use quite simple unambiguous definitions of invention and discovery: -- "invention" is the creation of something that didn't exi...

The wheel is an invention related to the discovery of some physical facts of nature: the rolling friction of a round body is much smaller that the cre...

I think the answer is YES, and I think there should be an objective way to distinguish if the regularities are due to the way we built our axioms or a...

I think the fundamental reason is that some properties of elementary particles (described by quantum mechanics) are intrinsically "discrete", whereas ...

. I agree. But do you think is possible to give a concrete meaning (or measure) to what it means for a theorem to be "beautiful"? I mean: if a large g...

I agree with you when you say that . So, I think that the concepts of numbers and geometrical objects are in some sense related to the physics of our ...