“Mathematics has no generally accepted definition.”
Interesting. I was surprised not to find the working definition I use listed among the main categories. I view mathematics as abstract magic - the art of those things unseen which can be manipulated and used as real providing only they are named, and thus invoked, precisely enough; their discovery or creation, naming, and manipulation.
It studies the structure of the constructs of the human minds that can be comunicated through language.
That's why we find so much math in #Science, because Science is our attempt to force our perception of reality into such structure.
It's not science until we can describe it with math, moving it from one person mind to the humanity culture.
I'd argue that mathematics of different species would converge.
From my POV, mathematics is about finding interesting patterns, extracting them from the phenomena that exhibit them, finding ways they can be further generalized or extended, and studying the properties of those patterns which are independent of where these patterns appear.
Look at the words you use to realize how human is Mathematics: "interesting" "patterns" "generalized" "extended" "studying"
The mathematics of different species COULD converge, but they would need to find a way to communicate mental constructs that might be completely incompatible.
Math is not about the patterns that exist, it's about the patterns that we can perceive or conceive.
But we perceive the pattern because they exist.
If there were no patterns in the world, i.e. if every time you do something it has unpredictable consequences, then no creature could develop any means of increasing its chance for survival.
I agree that "interesting" is a subjective word, but it only decides which mathematical concepts we discover. If 2 species find the same pattern interesting, they'll end up discovering the same abstract concept behind them.
Also, it's not like being interested itself is something unique to humans. Many other species also exhibit curiosity.
I think that if we found a different civilization whose development level is in the same order of magnitude as ours, they'd also have concepts like "big" and "small", natural numbers, and probably some understanding of "analogy".
@Shamar @Azure @alcinnz @freakazoid @feonixrift @natecull @ondiz @Wolf480pl @aral So much of math is just about consistency that rather than defining mathematics in terms of magnitude or cardinality, it makes more sense to define it as the study of consistency. Math is full of counterfactuals, so "the study of consistent things that are true" is wrong, but magnitude is totally irrelevant to (say) symbolic logic.
They strive to use a common language. They want to learn (by the very definition of mathematician) so the strive to understand each other and improve their language.
But it's always a HUMAN language.
A language that evolve, like humans and like math. But a language. Many specialized languages, actually.
@Shamar I think we're touching on a topic that appeared in "The Cambridge Quintet", where Turing insisted that a Turing Machine can do any kind of reasoning that a human can do, while someone else (Wittgenstein?) argued that without having human-like sensory experience, the machine will never understand the semantics of the language.
IMO we're all just turing machines.
(Given enough paper and ink. Otherwise, we're just finite state automata.)
I wonder why a human might want to describe himself as a machine.
Do you percieve those things in your head? Those ideas?
Machines do not have them.
Anyway, I'm not saying that a theoretical alien/artificial intelligence could not possibly understand the semantics of a human language.
I'm saying that Math studies the ideas that humans minds can conceive and share through language.
>Machines do not have them.
IMO there's nothing preventing them from having them.
>why a human might want to describe himself as a machine.
Maybe because I'm humble.
Certainly because I don't think I'm fundamentally any better than the computer I'm using right now.
Because I don't see a reason why a bunch of neurons connected together would fundamentally be any smarter than a bunch of transistors connected together.
@Shamar @Azure @alcinnz @freakazoid @feonixrift @natecull @ondiz @Wolf480pl @aral Sure, there's a reality. And, sure, we don't percieve it. But neither is mathematics particularly good about illuminating it -- mathematics merely illuminates different parts than our senses do. (Consider the banach-tarsky paradox.)
The only part of reality that Mathematics really "illuminate" is how our specie reason, how our mind ended up working in 4 milions years.
That's why I find very amusing that Math can be used to study Math.
Numbers, Vectors, Shapes, Predicates... are just constructs of our minds.
It's the best tool we have, but we should not fool ourselves about what it is.
We never explain anything with math. We describe it.
@Shamar @aral @Wolf480pl @ondiz @natecull @feonixrift @alcinnz @Azure @enkiv2 I'm not sure I understand this discussion. Any technological species is going to have a concept of number. They will also have a concept of zero, negative numbers, complex and hypercomplex numbers. They'll have polynomials, sets, and things like vectors, matrices, and tensors. These are universal, not human. They may express them differently, but they will have equivalents of each of these objects in their maths.
@freakazoid @enkiv2 @Azure @alcinnz @natecull @ondiz @Wolf480pl @aral @Shamar If math is exploration of an external reality, there is still a great deal of choice in the paths explored. There may be entire types of paths which one culture (or species) sees and another does not. Perhaps, metaphorically, we walk and they swim.
Math study how we describe what we explore.
Thus it's human in how we perceive, how our brain reason and how we describe.
Now, it's pretty obvious that it looks universal because wherever our brains look, in every direction, they always see through the same lens.
Math study the structure of such lens of our mind.
@Shamar @aral @Wolf480pl @ondiz @natecull @alcinnz @Azure @enkiv2 @feonixrift In David Brin's Uplift series, the Galactics didn't have symbolic calculus because they'd had computers for so long; they solved everything with numerical methods. He's a physicist, so he'd probably know, but it seems like you still need to be able to at least write down differential equations. I guess if you always used numerical solvers you might not bother with symbolic algebra either.
@kragen @Shamar @aral @Wolf480pl @ondiz @natecull @alcinnz @Azure @enkiv2 @feonixrift Having a way to write down equations is different from having techniques to transform one equation into another or directly solve the equation other than by brute force.
The space of problems which cannot be solved symbolically is dramatically larger than the space of problems which can, so you need numerical methods no matter what. You don't need symbolic methods AFAICT.
@freakazoid @feonixrift @enkiv2 @Azure @alcinnz @natecull @ondiz @Wolf480pl @aral @Shamar Without techniques to transform one equation into another, you can't show that two statements of the problem are equivalent, or that a solver that can solve one is applicable to the other. And you don't have to solve a problem completely in order to make a brute-force computation many orders of magnitude more efficient.
@freakazoid @enkiv2 @Azure @alcinnz @natecull @ondiz @Wolf480pl @aral @Shamar @kragen In some cases a symbolic form becomes shorthand for objects representable as a result of a numerical method. My current favorite such are 'formal' power series in p-adic representations - they look and act just like power series, and 'converge' but not to a number. However even decimal expansion of real numbers could qualify as such, formalized by Dedekind cuts.
Maybe, a few millions years from now, we will build or met an intelligent machines with free will and ethics.
We could also build or met unicorns, in few millions years!
But CS is as near to build an AGI as aerospace engineering is near to reach the Andromeda Galaxy.
@freakazoid @feonixrift @enkiv2 @alcinnz @natecull @ondiz @Wolf480pl @aral @Shamar You may be interested in Hartry Field's Science without Numbers. It reconstructs all of Newtonian mechanics in terms of points in space and betweenness. While it makes a philosophical argument I don't agree with, it's absolutely wonderful and awesome fun.
This is VERY interesting.
I know nothing about nominalism.
It seems pretty related to my insight but still a bit offset. As far as I can read, Fields say that the existence (the truth) of Math's abstractions is irrelevant to nature.
I say that the whole Math is irrelevant to nature, that it's all about our minds.
It's like if we could only observe the world as reflected into a mirror.
That mirror is our mind.
It has a shape and it has colours and holes that inevitably deform the image we see.
But since we cannot remove the mirror, since we see through mirror from our births, we think it's structure is universal.
Mathematics study the structure of the mirror that looks the same to every human, and thus deform reality in the same way for each of us.
The part of the mirror we can talk about, even if we are not aware it's just a mirror.
What my definition say is: hey, Math is beautiful because WE are beautiful.
It's inside us!
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