I hardly understand anything in this thread, as my knowledge of mathematics is rudimentary. — Wayfarer
If you point to a number, '7', what you're indicating is a symbol, whereas the number itself is an intellectual act. And furthermore, it is an intellectual act which is the same for all who can count. It's a very simple point, but I think it has profound implications — Wayfarer
My view is that numbers are real, but not physically existent. — Wayfarer
If you point to a number, '7', what you're indicating is a symbol, whereas the number itself is an intellectual act. — Wayfarer
And furthermore, it is an intellectual act which is the same for all who can count. — Wayfarer
It is (ZF\I)+~I that is bi-interpretable with PA. — TonesInDeepFreeze
So you don't accept that 7=7? — Wayfarer
https://victoriagitman.github.io/talks/2015/04/22/an-introduction-to-nonstandard-model-of-arithmetic.html
In particular, a nonstandard model of arithmetic can have indiscernible numbers that share all the same properties.
R. Kossak and J. H. Schmerl, The structure of models of Peano arithmetic, vol. 50.
I agree, also with Yanofsky.My intuition says that Yanofsky is probably right, and that it is Cantor's theorem that is at the root of it all, but I am currently still struggling with the details of what he writes. — Tarskian
If you try to express all the truth about the natural numbers, you are effectively trying to create an onto mapping between the natural numbers and its power set, the real numbers, in violation of Cantor's theorem.
Any observations on the arguments for or against mathematical platonism as outlined in this post? — Wayfarer
https://en.m.wikipedia.org/wiki/Philosophy_of_mathematics
Davis and Hersh have suggested in their 1999 book The Mathematical Experience that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism.
Kurt Gödel's Platonism postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly.
You can simply switch off the lights and declare that it is all just meaningless symbol manipulation and about nothing at all, which it actually is, if you take the time to think about it. — Tarskian
To reify is to 'make into a thing'. Numbers don't exist as objects, except for in the metaphorical sense of 'objects of thought'. — Wayfarer
Except it doesn’t allow for the iunreasonable effectiveness of mathematics in the natural sciences. — Wayfarer
As I said above, the reason the most people won’t defend platonism is because they don’t understand or can’t live with the metaphysical commitment it entails. Myself, I have no such difficulty. — Wayfarer
https://en.wikipedia.org/wiki/Pythagoreanism
Pythagoras, in his teachings focused on the significance of numerology, he believed that numbers themselves explained the true nature of the Universe. Numbers were in the Greek world of Pythagoras' days natural numbers – that is positive integers (there was no zero).
In modern lingo, arithmetical theory, i.e. the theory of the natural numbers (PA), and the unknown theory of the physical universe exhibit important model-theoretical similarities.
For example, the arithmetical universe is part of a multiverse. I am convinced that the physical universe is also part of a multiverse.
The metaphysics of the physical universe is in my opinion nothing else than its model theory.
Model theory pushes you into a very Platonic mode of looking at things. In my opinion, it is not even possible to understand model theory without Platonically interpreting what it says. — Tarskian
If we don't differentiate between objects sensed and ideas grasped by the intellect. then there is nothing to prevent us from believing that the universe is composed of numbers. This is known as Pythagorean idealism, and often called Platonism. — Metaphysician Undercover
https://en.m.wikipedia.org/wiki/Map%E2%80%93territory_relation
The map–territory relation is the relationship between an object and a representation of that object, as in the relation between a geographical territory and a map of it. Mistaking the map for the territory is a logical fallacy that occurs when someone confuses the semantics of a term with what it represents. Polish-American scientist and philosopher Alfred Korzybski remarked that "the map is not the territory" and that "the word is not the thing", encapsulating his view that an abstraction derived from something, or a reaction to it, is not the thing itself. Korzybski held that many people do confuse maps with territories, that is, confuse conceptual models of reality with reality itself.
For example, if there are five people in a group, this situation is structurally similar to a set with five numbers. It does not mean that a person would be a number. — Tarskian
You could conceivably make a digital simulation of the entire universe and run it on a computer. This simulation of the universe would consist of just numbers. What you would see on the screen will be an exact replica of what you would see in the physical world. It would still not mean that this collection of numbers would be the universe itself. — Tarskian
A map of the world can help us understand the world. The map will, however, never be the world itself. — Tarskian
Now, if it is about an abstract world, then the perfect map of such abstract world is indeed the abstract world itself. There is no difference between a perfect simulation of an abstract world and the abstract world itself. — Tarskian
It's structurally similar because what constitutes "a group" is artificial, just like what constitutes "a set" is artificial. So you are just comparing two human compositions, the conception of a group and the conception of a set.. — Metaphysician Undercover
How would your proposed computer simulation provide a "better" replica of the universe? — Metaphysician Undercover
what would make an abstract world the perfect abstract world? Do you see what I mean? — Metaphysician Undercover
https://en.m.wikipedia.org/wiki/Isomorphism
The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism.
https://en.m.wikipedia.org/wiki/Law_of_identity
Wilhelm Wundt credits Gottfried Leibniz with the symbolic formulation, "A is A".[4] Leibniz's Law is a similar principle, that if two objects have all the same properties, they are in fact one and the same.
Nor do I lean back and say, Wow, that's true! I simply don't use the words "true" or "truth" when doing math. I don't even think the words. But that's me, not other math people. — jgill
I don't think of myself doing anything. I only do. Or did. I'm pretty old and not in such great shape to do much of anything. — jgill
Doesn't surprise me. I am (was) a humble classical analysis drone, far from more modern and more abstract topics. Maybe young math profs these days use the word "truth" frequently. — jgill
(On the other hand I did point out what I considered the truth of a form of rock climbing many years ago by demonstrating and encouraging a more athletic, gymnastic perception of the sport. Even then I didn't use the word "truth".) — jgill
Did you think your work was "about" anything? Or pure symbol-pushing?
I'm pressing you on this point because I don't believe you did not believe in the things you were studying! — fishfry
Gravity is true, wouldn't you say? — fishfry
I never spent any time thinking about what I was doing. I did it, and still do it because it is a fascinating realm of exploration. As was rock climbing when I was a lot younger. I never puzzled over the fundamental nature of mathematics. And I doubt my colleagues did either. — jgill
No. Gravity simply is. Some aspects could be said to be true. Word babble IMO. — jgill
Hence Wheeler’s conjecture of the One Electron Universe — Wayfarer
proposed by theoretical physicist John Wheeler in a telephone call to Richard Feynman in the spring of 1940.
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