“Mathematics has no generally accepted definition.”

en.m.wikipedia.org/wiki/Defini

@aral
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.

@aral
(Of which, in the typical manner of magic, there are entire worlds. Each of which plays by its own very peculiar rules.)

: μάθημα (máthema) τικὴ (tikḗ, tékhnē)
is "the art of teaching" and "the art of learning".

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 , 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.

@Shamar What did I do? 😂 Great piece btw, very useful for when people ask me if I'm a scientist because I'm an engineer 😀

@Shamar Maybe when we talked about your talk? 🤔 (I took only one coffee today so don't trust me)

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.

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.

@Shamar @Azure @alcinnz @freakazoid @feonixrift @natecull @ondiz @Wolf480pl @aral Turns out consistent things are useful mental tools even if they don't really apply to our universe.

Well yeah, magnitude is irrelevant in many fields of math. But what about partial order? Logic has the implication, set theory has subset-of.

@Shamar

yet many of the creatures perceive the reality in a similar way to ours, at least at the lowers level: they have sight, smell, hearing.

But even without that, I'm pretty sure that they would have natural numbers anyway.

@Shamar well yeah, language is the foundation of all formal reasoning. Especially written language.

@Shamar yet I think how a particular language is implemented will have little influence over the maths you discover.

@Shamar then how come human mathematicians all over the world, speaking different languages, use the same language for their mathematical expressions?

@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.)

>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.

Anyway, I don't think I can convince you, and neither can you convince me. So let's just agree to disagree.

@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.)

@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.

@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.

@freakazoid @feonixrift @enkiv2 @Azure @alcinnz @natecull @ondiz @Wolf480pl @aral @Shamar I'm skeptical about this idea. How do you think about how well different solvers might work on the same problem if you don't have a way to write down the problem?

@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.

@feonixrift @enkiv2 @Azure @alcinnz @natecull @ondiz @Wolf480pl @aral @Shamar @kragen It does seem unlikely, though, that nobody would notice that certain solutions looked exactly like other solutions and develop methods to figure out what those are in the absence of such methods.

@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.

@kragen @Shamar @aral @Wolf480pl @ondiz @natecull @alcinnz @Azure @enkiv2 @feonixrift Brin was talking about species that had never in their existence not had virtually limitless computing power. You could think of it as post-scarcity computing.

@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.

@Shamar @aral @Wolf480pl @ondiz @natecull @alcinnz @Azure @enkiv2 @feonixrift @kragen I don't know, machines might end up doing a much better job at ethics than humans do.

@Shamar @freakazoid @feonixrift @enkiv2 @Azure @alcinnz @natecull @ondiz @Wolf480pl @aral We know how far away Andromeda is. We don't have any idea how far way AGI is.

@Shamar @aral @Wolf480pl @ondiz @natecull @alcinnz @Azure @enkiv2 @feonixrift @kragen I don't think that having computers significantly improve how we make ethical decisions necessarily require AGI. Whether that really qualifies as "teaching computers ethics" I don't know, since "teaching" can be interpreted lots of different ways. But humans can barely do ethics, just like we can barely drive cars.

@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.

@Shamar @I disagree with that. It can be science without using math. It's just that scientific principles are easiest to express in a language designed for it. Karl Popper, who wrote the book on the scientific method, didn't write it in maths. He wrote it in philosophy.

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