I love articles like this. Thanks for sharing. Ultimately math is just a language invented to describe the world. Which is what any language does. And it's a really useful language for sharing certain kinds of intelligence. I got into a discussion with my son last night about sine waves. Which don't really exist, per se, but they are a useful language construct for thinking about waves (especially sound waves as we were discussing). Shared intelligence like this is, as some have maintained, a major differentiator between humans and pre-humans. So the very fact we can argue about what is a number is part of what it is to be human.
What Even Is a Number?
21–30 of 173 posts
Re: What Even Is a Number?
#22I 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…
Re: What Even Is a Number?
#23I 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 matter/energy/space/time is actually THE number two.)
To me, it's wonderful to reflect on the epistemological status of numbers. They don't seem to exist, but we talk about them as if they do. Certainly they are useful. But they don't exist any more than, say, Sherlock Holmes does.
It's also much fun to reflect that, whatever the epistemological status of number, that status is shared by computer languages and algorithms and much else. E.g. the C language doesn't exist. The C standard(s) exist, many C compilers exist, lots and lots of C code exist, etc., but the language itself doesn't exist anywhere, any more than does the number two.
Re: What Even Is a Number?
#24I 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…
That's just "Exist as physical objects exist" Ie., you're asking someone to find you an object with spatio-temporal properties which, as an object, does not have spatio-temporal properties. Likewise you could ask, "find me a smile?!!" and then deny every example was a smile because "it was only a face smiling!". "Existence" is defined relative to a context in which it can be evaluated, and it -- in general -- does no…
Yes, I could have really tortured my kids and said, you can't even show me a "ball", because it will always be a specific ball and not the archetypal concept of "ball".
I probably should, it'd be fun and they're a bit older than when I pulled the "one" game.
Re: What Even Is a Number?
#25I 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…
That's just "Exist as physical objects exist" Ie., you're asking someone to find you an object with spatio-temporal properties which, as an object, does not have spatio-temporal properties. Likewise you could ask, "find me a smile?!!" and then deny every example was a smile because "it was only a face smiling!". "Existence" is defined relative to a context in which it can be evaluated, and it -- in general -- does no…
I believe that in fact the numbers do exist as physical objects exist. Just in their own universe.
When we reason about numbers, we're embedding a representation of a universe of numbers into our universe.
What we call "physical existence" could is also probably just be "math all the way down". Just not in such a way that the number two per se can be an entity for us to behold.
Re: What Even Is a Number?
#26> You can see some of that history of what we call some of the different sorts of numbers: e.g. Negative, irrational, and imaginary numbers. Each of these represents a bitter argument about whether a new type of number should be allowed, all of which were eventually won by the people on the side of the new numbers... Was there ever a proposed type of number that is today not considered a number? Like i=squareroot(-1)…
The original version of Calculus by Newton used "fluxions". That doesn't correspond to any number system that we use today.
Leibniz's (re?)invention of Calculus used infinitesmals. Infinitesmals as understood by mathematicians then do not correspond to any numbers we use today. (Yes, yes, something else called infinitesmals do show up in nonstandard analysis and the notation deliberately looks the same. But the underlying concepts are more..complicated.)
Re: What Even Is a Number?
#27There is a video (https://www.youtube.com/watch?v=S4zfmcTC5bM) from the PBS Infinite Series that covered this topic as well, for people that what a visual version of the explanation. I'm still sad that the series/channel was shutdown though, such a great series.
PS: this article/video essentially defines the naturals from the fundamental set theory axioms. This other video (https://www.youtube.com/watch?v=KTUVdXI2vng) from the same PBS series shows how you can then use the naturals to construct other types of common numbers, up to the reals.
Re: What Even Is a Number?
#28Article seems very verbose and only really addresses the natural numbers (it mentions negatives and rationals in passing). In case anyone just wants actual answers without reading pages and pages of prose, here is one set of constructions (there are other competitors too). The natural 0 is the emptyset. The natural 1 is the singleton {0}. The natural 2 is {0,1}. In general, the natural n+1 is {0,...,n}. Exercise to t…
Your castle is built on sand. What, then, is {}?
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Re: What Even Is a Number?
#29Article seems very verbose and only really addresses the natural numbers (it mentions negatives and rationals in passing). In case anyone just wants actual answers without reading pages and pages of prose, here is one set of constructions (there are other competitors too). The natural 0 is the emptyset. The natural 1 is the singleton {0}. The natural 2 is {0,1}. In general, the natural n+1 is {0,...,n}. Exercise to t…
Your castle is built on sand. What, then, is {}?
{} is the empty set, which as mentioned is natural 0.
Re: What Even Is a Number?
#30Article seems very verbose and only really addresses the natural numbers (it mentions negatives and rationals in passing). In case anyone just wants actual answers without reading pages and pages of prose, here is one set of constructions (there are other competitors too). The natural 0 is the emptyset. The natural 1 is the singleton {0}. The natural 2 is {0,1}. In general, the natural n+1 is {0,...,n}. Exercise to t…
”an equivalent relation.”
Auto-correct at work, I guess. For those not familiar with the term: if Google says did you mean equivalence relation, Google is right.