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What Even Is a Number?

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Re: What Even Is a Number?

#121

Post by FabHK »

kazinator wrote: post #105
FabHK wrote: post #98

Earlier quoted context omitted.

> How is it any different than believing that unicorns exist as physical objects, just in their own universe? Good question, but I think there might be a difference. Suppose, in a galaxy far, far away, there are aliens. They probably don't have the concept of unicorns. However, they probably have discovered numbers, pi, Pythagoras theorem, e^(i pi) + 1 = 0, etc. So, does the fact that something is discovered (rather…

I would sadly have to say no. Because in some completely different universe unlike ours, the aliens have no concept of, say, "hemoglobin" because there isn't matter like ours: no atoms, no iron, no carbon. Unlike numbers, hemoglobin and unicorns contain a kind of arbitrary complexity. Yet one is definitely real. Some real entities are far removed from elementary math (even though they may ultimately rest upon it, man…

I'm not claiming that if something exists (haemoglobin), we must have a notion of it and true statements about it in every universe.

Rather: if we have a notion of something and statements about it in every universe, then it must exist. However, come to think of it, that's bonkers.

Still, it's remarkable that mathematics allows us to make reproducible true statements about fictional objects.

Re: What Even Is a Number?

#122

Post by ngcc_hk »

I think the counting is natural to human. The corresponding theory op used is not natural. If one has to use that, one start to use writing to do correspondence. Or use shell/good/coin.

That is for +integer.

Not sure about how natural is real number other than pi using geometry. And irrational also use geometry.

Use algebra and group-ring-field is very late.

Re: What Even Is a Number?

#123

Post by samhain »

cobbzilla wrote: post #10

I had a fun mind-play with my kids; I asked them if numbers like "one" and "two" really exist. They said yes, of course. OK, I said, show me a plain old "one". Nope, that's one fork; nope that's one ball; nope, you get the idea. You cannot show just "one" unattached to anything else. The concept of "one" is an idea. It only exists as a "real" concept in your mind because it exists as the same concept in the minds of…

So by your definition, one is an adjective, so asking them for a noun that represents the adjective "one" violates the definition. It's sort of a trivially false disingenuous question.

Re: What Even Is a Number?

#124

Post by Retra »

roywiggins wrote: post #104
Retra wrote: post #88

Earlier quoted context omitted.

That other "universe" is just patterns embedded in our brains. That's all you need. You could likewise argue that sounds exist in a different universe, but are only "embedded" into the air using pressure waves, but why would you ever think you needed a different universe of sounds when you've already got pressure waves in air? I mean... what's "not enough" about that for you? >What we call "physical existence" could…

The strange thing about mathematics, as opposed to other noncorporeal ideas, is that it is nearly uniquely suited to being transmitted between brains. My idea of the number "three" and your idea the number "three" are probably very similar. This sort sensation that they have an independent existence leads people to the Platonic notion of a mathematical "realm" or "universe". (personally I think mathematics exists in…

You could say that about most language we use: it's meant for transmitting thoughts between human brains. Why would it be bad at it?

Besides, there are plenty of mathematical concepts which are hard to understand. I don't see any uniqueness to math here.

Re: What Even Is a Number?

#125

Post by Retra »

kazinator wrote: post #109
Retra wrote: post #88

Earlier quoted context omitted.

That other "universe" is just patterns embedded in our brains. That's all you need. You could likewise argue that sounds exist in a different universe, but are only "embedded" into the air using pressure waves, but why would you ever think you needed a different universe of sounds when you've already got pressure waves in air? I mean... what's "not enough" about that for you? >What we call "physical existence" could…

> abject poetic nonsense. No it isn't! Either it is math all the way down, or else it isn't. If it isn't, then what is at the bottom? So this is actually an existential question. Suppose we reach a final description of reality that is entirely accurate (no longer an approximation that fails to account for some structures and processes). Won't that description consist of nothing but math? Then, if it is complete, how…

>Suppose we reach a final description of reality that is entirely accurate (no longer an approximation that fails to account for some structures and processes).

> Won't that description consist of nothing but math?

Math is just words. Any description of anything can be reduced to math if you assume the proper math will be developed to describe it. But the same could be said of any subject. You could say the universe is Mandarin Chinese if you find a perfect description in that language. You could say the universe is binary code if you found a perfect description of it in binary. You could say the universe is tree bark if you found a perfect description of it in tree bark.

You're saying nothing useful. You're just confusing the map for the territory. (And arguably, you will never find a complete description of the entire universe that fits in a network of human brains, so the universe can't be math for that reason alone.)

Re: What Even Is a Number?

#126

Post by olooney »

carapace wrote: post #23
cobbzilla wrote: post #10

I had a fun mind-play with my kids; I asked them if numbers like "one" and "two" really exist. They said yes, of course. OK, I said, show me a plain old "one". Nope, that's one fork; nope that's one ball; nope, you get the idea. You cannot show just "one" unattached to anything else. The concept of "one" is an idea. It only exists as a "real" concept in your mind because it exists as the same concept in the minds of…

Ah! I wrote the below before seeing your comment: Numbers don't exist. Take the number two. You can have two apples but that's not the number two. You can write the numeral "2", but that's not the number two. It's one line. The word "two" has three letters, it's obviously not the number two. In fact, the number two doesn't exist (or it's existence is not contingent on an arrangement of matter/energy. No pattern of ma…

Proof that at least one number exists: Take for our logic first order logic with equality[1]. Take for our definition of the word "number" the Peano axioms[2]. The first Peano axiom is "0 is a natural number." Define "N" to be the unary relation "is a natural number." In our formal language, we write the first Peano axiom as "N0". By the principle of existential generalization[3], we have "∃x Nx", which we read as "there exists x such that x is a natural number." Q.E.D.

I assume you find the above deeply unsatisfying. Most people do. Yet I find it clarifies the matter entirely! If a person's conception of "number" is consistent with the Peano axioms, and if that person's use of the word "exists" includes the existential logical quantifier, then for that person "numbers exist" is a theorem. If a person uses other definitions, the matter may be otherwise.

[1]: https://plato.stanford.edu/entries/logic-classical/ [2]: https://en.wikipedia.org/wiki/Peano_axioms [3]: https://en.wikipedia.org/wiki/Existential_generalization

Re: What Even Is a Number?

#127

Post by aidenn0 »

The article ends with "Almost certainly not, but if so why would we care?" which I find (along with its inverse "Almost certainly, but if not why would we care") to be my answer to many philosophical questions.

Re: What Even Is a Number?

#128

Post by ngcc_hk »

dragonwriter wrote: post #111
stcredzero wrote: post #106

Earlier quoted context omitted.

However, they probably have discovered numbers, pi, Pythagoras theorem, e^(i pi) + 1 = 0, etc. We can only be confident of that if they're in this universe. Also, this is only a conjecture. It's just something some of us imagine would be true. You can't elevate such an idea to the level of empirical data. So, does the fact that something is discovered (rather than invented), and can be discovered many times independe…

> Unicorns. Kirin. Narwhals. Rhinos. So you have two independent origins in culture for such a creature, plus two appearances of something similar in nature. It's not at all clear that the cultural constructs are independent of each other or real world creatures. Particularly in the Western case, some of the earliest Greek accounts of the unicorn as well as Marco Polo’s account confirming its existence and difference…

Kirin is giraffe and upset Confucius as it should not appear in his chaotic time.

Re: What Even Is a Number?

#129

Post by qubex »

WilliamEdward wrote: post #119
qubex wrote: post #101

Earlier quoted context omitted.

I’m a mathematician and you seem a bit confused. Natural numbers ({1, 2, 3, ... }) are for counting . Giving a bunch of objects as being prime examples of that which can be counted with natural numbers is as close as one can get (short of just scribbling down the numerals and kind-of circumventing the point). Other classes of numbers (particularly the Reals) are for measuring . Again, short of writing down examples o…

I think they're just trying to say "1" is not a tangible object in the physical universe, not that it does not exist as a concept. The natural numbers are just data, information, they don't exist beyond the first dimension except as descriptors to things in further dimensions, just like lines can be built up into higher dimensions but cannot be physically created for us to observe. Maybe you're reading too much into…

How is two straight lines crossing in one and only one place not an example of a single occurrence without reference to real world objects (which the OP dismissed as “not really being a number”).

As for reading “too much into this”, as Instated: Irma a mathematician, of course I’m going to take this seriously and make an effort to disprove naive misconceptions.

Re: What Even Is a Number?

#130

Post by qubex »

Nomentatus wrote: post #67
cobbzilla wrote: post #10

I had a fun mind-play with my kids; I asked them if numbers like "one" and "two" really exist. They said yes, of course. OK, I said, show me a plain old "one". Nope, that's one fork; nope that's one ball; nope, you get the idea. You cannot show just "one" unattached to anything else. The concept of "one" is an idea. It only exists as a "real" concept in your mind because it exists as the same concept in the minds of…

"Reification" is the term to search for if you wish to delve into the extensive Philosophical bibliography on this topic that's accumulated over the last few centuries. Within mathematics "Intuitionist Mathematics" is the rubric for the anti-reification view. This relies on computability as a substitute for "existence." The Intuitionist view has pretty much triumphed.

I’m all for reification. It’s not even limited to mathematics: ‘democracy’ is just another example of a reification from another realm of human endeavor: it gets implemented in different countries with slightly different axioms and varying degrees of success but in aggregate those nations that abide by some form of it are accepted as functional equals by other whom have likewise implemented their own instantiation of the overarching ideal.
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