Gregory Chaitin - Is Mathematics Invented or Discovered? 

C Oda yes

I dont mean to be so off topic but does anyone know of a tool to get back into an instagram account? I somehow forgot my login password. I love any tricks you can give me

@Melvin Thaddeus instablaster =)

@Clayton Manuel i really appreciate your reply. I got to the site thru google and I'm trying it out atm. Seems to take quite some time so I will get back to you later with my results.

@@melvinthaddeus2943 I would presume that without any species that are organisms evolved into having reflective minds,there would be no mathematics.

He speaks wonderfully!! And writes even better!

@@LastSonofEther what have u read that you like?

@@Ensource i have read metamaths and currently reading unraveling complexity

Facts and London REAL

I was jut going to mention Wolfram as well. Chaitins view is very reasonable. There is pure math like calculus. But reality is extremely complex, perhaps derived from fractal like equations which we are unable to find or prove. Bu using computers we can emulate the same behavior. So its not exact an pure math but very good approximations. Pragmatism!

The platonic universe causes difficulties from the get go. Obviously it would be a kind of concept space. Among other things it might be too big. We talk of the class of all sets being too big but it would have to be part of such a concept space as a single unit. It is better of think of mathematics as linguistic conventions with the entailments that follow from definitions - basically a language game The question people are really interested in is mathematical realism. The question may be more pointedly formulated in this way. Does one think that every meaningful statement about the natural numbers(some closed formula in some logical calculus) is either TRUE or FALSE? The mathematician is allowed to define TRUE and FALSE as he pleases as long as TRUE and FALSE are mutually exclusive and collectively exhaustive. A positive commitment means you are a mathematical realist, and a refusal to commit means you are not. I think that will boil down to whether one believes in the natural numbers, whether one believes in the consistency of the Peano Axioms(which we cannot prove by Godel's second Incompleteness theorem) I think that all mathematical claims can be reduced to claims about the natural numbers. So that form of the question might cover everything. As far as calling human mathematics a historical artefact as Wolfram does, Wolfram talks of other axiom systems but even within ZFC, there's an immense deal(an infinite deal) we don't do that we could do. The choice of problems to work on is of course a historical artefact. And the arbitrariness of the axioms we use is not in itself very important when you consider that the claim of mathematics is that such axioms lead to such theorems, never the unconditional truth of the theorems and that is a truth that transcends any specific choice of axioms. So while we choose what problems to work on, the results themselves are absolute in that sense

Which God?

@@ASLUHLUHC3 Thee One and Only, YHWH, who is a Triune Deity. Father, Son (Jesus Christ), and Holy Spirit. You know Him?

One has to define a set to count the heads of cattle, and this set is arbitrary, then an integer depends on one's decision, it is not a part of Nature.

@@massecl a way to define numbers is to define a number as the class of all sets with a certain cardinality. In that way, natural numbers(and infinities btw) become the equivalence classes of the class of all sets under the equivalence relation of having the same cardinality. Intuitively that would connect the abstract notion of a natural number with Everyday experience, as we associate a set of 5 cows with the number 5 for example.

@@dekippiesip Defining a number by "same cardinality" is circular, just replacing the word. But we can define "same cardinality" by the existence of a bijection. Then the number again depends on the equivalence class, and this class is arbitrary. The number doesn't exist out there on its own, for there is no way out there to decide which class is the case. We may count the heads of cattle, but perhaps the "real world" counts the number of asparagine molecules or of bristles.

@@massecl yes that was where i'm getting at, define 'same cardinality' by bijection(no mentioning of 'numbers' then) and associate every number with an equivalence class under the equivalence relation. This class will then contain every possible set you can think of with that cardinality, including those you mentioned. The only issue could be Russels paradox, since we are sort of appealing to the 'set of all sets'. By calling it the 'class' of all sets I am trying to dodge that, but it may be a cheap attempt.

Integers may be inspired by nature but clearly not abstracted therefrom. Nobody's seen infinite sets or even very large finite numbers. And if math is real(meaning roughly that the Peano axioms are consistent, which we cannot prove courtesy Godel) then clearly this is reality that transcends physical reality - mathematical reality would hold in all possible worlds. But of course it remains an open question whether the Peano Axioms are indeed consistent

That would be interesting. I'm not sure why I do mathematics or find certain branches interesting

@@JM-us3fr Main main interest here is that it seems that proof theory cannot yet establish (even approximate) upper bounds for the proof length for a given unproven statement. So where does a mathematicians' intuition come from and does it work? Is it just trial and error? Does the platonic realm have enough structure to estimate "distances" to other objects?

Nobody said "God's work" and "Mind of God" is a figure of speech for a certain kind of concept space. This isn't religious in any sense you are thinking of

@@jamespower5165 It's been a couple of years, but I doubt that I was bringing anything religious into it, only that he was full of his own self importance, which he then projected onto his work. However, in these two years, I have begun to wonder if most advanced math wasn't "telling a story" rather than finding a mathematical basis for observations. This leads people to wonder about the uncanny accuracy of mathematics, when in reality it is just a story told with a different language.

​@@jamespower5165These things are not that different.

@blindwillie99 Ok, I have added this to my list of books I need to read. It is first in, first out at position 50, so it may take me a while to get back to you.

this is a modern age version of "god". When somebody claims that something exists in the physical world, we now require some evidence of that something in the physical world.

At least math has a firm footing in reality and is useful.

Bingo. We are God! HEAVEN is NOW

@@user-js2dr9gv1u Show me the transitive property in nature... in snowdrops. [QED]

As far as I know, even Pi can be subject to change in non-euclidian space or some other realities

This is just the point: the mathematician gets to decide what to take as axiomatic, and see where it leads. Isn’t this just what Riemann and Lobachevski did with Euclid’s parallel postulate? Where’s the problem? On the other hand, Fermat’s Theorem and the four-color mapping conjecture eventually did end up proven as theorems, so who’s to say?

Could you explain why it was not interesting?

I disagree.

Light is a wave and particle at the same time or maybe we don't have the theoretical instruments to explain it any other way. Maybe we don't yet have the word to describe how did we end up with math. Some talk about math itself as a language and if that may be the case the question can go deeper. Is language itself discovered or invented?

The KINGDOM of HEAVEN is WITHIN

@joseph_goebbels606 Physics is not the same as mathematics. When creating new mathematics, you are free to choose any axioms you want as long as they are consistent. To say that mathematics is discovered is a bit like saying that when an author writes a book he discovers the story he is writing

I feel the same. I feel that anything abstract requires a physical universe. Abstractions of any kind requiring some sort of substrate to symbolically work with, and maintain it. Every abstraction we use requires physical tools of some sort or another.

Discovered. Although we do invent ways of discovering the discoveries.

Normal doesn't mean correct. Look up the word normal and I think you will understand. Normal is so misused in conventional speech, as to be nearly meaningless, unless you provide a more complete explanation.

Darwinian evolution has been the accepted foundation of biology for over a century. Denying it is like denying gravity.

Except there is no "one" or "two" - they are concepts. You can have one cow, or one ball etc., but you can't have a "one".

@@CyberiusT If you are being obtuse, fine, if not, no wonder the world is so F--ked up....

@Calum Tatum Yes. Initially maths was useful like for military formations, property management, accounting and calendar management. Then slowly form were abstracted. That sort of Mathematics was usefull. I still know my logarithms and matrices😀😀. But many of the more abstract fields of maths are not of immediate or even future use. The are just discoveries in a platonic land.

@Calum Tatum Yes. Maths is about a sense of beauty among many other things. However, automation theorem proving by machines is quite ugly. But yes, human interest is driven by order, symmetry, elegance and beauty which may not have any practical uses

@Calum Tatum Yes true. For example, if you are used to workflow systems (business flow software. I am into software) in the end there seem to be only two primitives : graphs or tables. The choice of one of these will dictate many aspects of others.

If a new "axiom" is proven to be a consequence of an axiomatic system, it is a theorem, not an axiom, by definition. This said, I agree (and had much the same thought) on the "risk of basing something on a non-intuitive axiom that appears to be true" - however the approach is used already: quite a few "proofs" in number theory assume the Riemann Hypothesis, which is far from proven in any mathematical sense, but seems to hold to any empirical test (as did Euler's conjecture on sums of powers... for about 200 years)

Your statement is the result of complete ignorance about logic.

@@schmetterling4477 No you.

Thats because spacetime doesnt ! The only explanation for space and time is that they have to be emergent . Its like that episod from Rick and Morty with Universes into universes ...and we cant never see whats outside of our own matter Universe . ;...just by imagination.

Mathematics has nothing to do with empiricism. Empiricism derives complex physical phenomena from simple ones. Mathematics merely makes up axioms and derives theorems from them using a pre-selected set of rules that can be changed arbitrarily.

Its more a question of math being a projection on the world, to help us understand it, as opposed to some 'twoness' existing as a state in the world. That is to say, our concept of moons, our dividing up the world, etc. being a sort of systematic language that we use (And create) to describe and reflect states of the world. But as he describes, pure math is seperate from the world.

@@alasdairmacintyre9383 This particular topic is a hard one for me. The question seems to fold back in on itself.

Truth is one of two possible logic values in one particular binary logic system (there are many more such systems that we just don't use much). You are free to construct logic systems with more than two values or even an infinity of values and people have. We just haven't spent all that much time on exploring whether that leads to interesting results or not.

go f... yourself you theist fuck

@Trevor Chase, don’t have a problem with axioms. Mathematical axioms still have intuitive basis for them. For example 2=2 doesn’t require proof as it is intuitively obvious. My problem is with axioms that require massive leaps of faith and from my readings about Godel he seems to have no problems with those.

Where do our thoughts come from? Are they real? 2+2=4 is empirical as well as factual and real.

Your reasoning is based on the false premise that mathematics (or anything else for that matter) is purely a representation of the physical world.

@@dlevi67 Ah yes - the original sin . Our spirits manifesting in the material realm - our Karmic fallout for being creatures of desire .

@@mulliniks51 That's one way of looking at it - not necessarily the only way. Let me rephrase my original point: "Your reasoning is based on the false premise that mathematics (or anything else for that matter) _must_ _be_ purely a representation of the physical world.