31–40 of 173 posts
Article 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…
Article 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…
For anyone who wants to learn more about this: what the parent comment is describing is the construction of natural numbers as cardinalities of sets. This is the (modified) Frege definition which avoids Russell's paradox - you can use that as a jumping off point for further dives into set theory and number systems.
To define Peano arithmetic you start with a successor function, then an addition function, then multiplication, etc.
It's a category. 1 is the category of singletons, 2 is the category of pairs, etc.
As to what numbers are, they're an ordered collection of categories.
Edit: To whoever downvoted this comment, kindly explain. Insofar as I'm aware this is textbook maths, psychology, and philosophy.
Article 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.
Earlier quoted context omitted.
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.
How is it any different than believing that unicorns exist as physical objects, just in their own universe?
What we call "physical existence" could...just be "math all the way down".
Or, math is also a number of simplified alternate views of reality of varying degrees of "applicability." (Where "applicability" has to do with predictive power over reality.)
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…
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…
> What even is a number? It's a category. 1 is the category of singletons, 2 is the category of pairs, etc. As to what numbers are, they're an ordered collection of categories. Edit: To whoever downvoted this comment, kindly explain. Insofar as I'm aware this is textbook maths, psychology, and philosophy.
But if you need to avoid Russell's paradox the sets defining each natural n can't contain n as an element. The easiest such construction was outlined elsewhere in this thread by xamael. Under your category theoretic construction, every natural number n is defined as the category of sets having cardinality n. But then every nth category will necessarily contain infinitely many sets with cardinality n that also contain n. That causes the paradox.
Unfortunately I don't think you can construct the naturals in a category theoretic way while avoiding Russell's paradox since any category by cardinality will fall into that trap. But if you don't need to mind that problem, this is neat.
Responding to your edit: I didn't downvote you; in fact I upvoted this comment because it's correct and it was gray at the time of my writing. My comment is just a point of clarification.
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…
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…
Only for the sense of "exist" which is roughly equivalent to "can poke a hypothetical indestructible stick at it, given enough energy."
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.
For the purpose of tax law, software is defined as a "tangible good." However, in other areas of the law, it's considered intangible.
