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What Even Is a Number?

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Re: What Even Is a Number?

#141

Post by mzs »

qubex wrote: post #137
mzs wrote: post #74

Earlier quoted context omitted.

Historically true, but glad it's there so ℤ is a group. The way to think about 0 for a child is what you add to anything so it stays the same, ie nothing.

Hate to be pedantic, but ℤ︎ is a set (not a group) and even with 0 adjoined it only forms a group under the operators of addition, subtraction, and multiplication.. If you want a set that also forms a group under division you need to ‘upgrade’ to the set ℚ︎.

ℤ︎ (…, -2, -1, 0, 1, 2, …) and + forms a group. It is also a ring with * and +. ℚ︎ with * and + forms a field.

Re: What Even Is a Number?

#142

Post by dragonwriter »

ngcc_hk wrote: post #128
dragonwriter wrote: post #111

Earlier quoted context omitted.

> 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.

Some accounts of Kirin, yes, appear quite likely to be giraffes, others are nearly identical to the Western accounts of the monoceros of distant India (one of the original of the unicorn legend) and are almost certainly accounts of rhinoceros.

Re: What Even Is a Number?

#143

Post by _7fvc »

I often hear people say mathematics is the language of science. Since learning about cryptocurrency, I came to realize that more precisely, mathematics is the language to describe energy transformation.

So a number can be seen as a metaphysical representation of energy.

Re: What Even Is a Number?

#144

Post by btilly »

FakeComments wrote: post #132
WilliamEdward wrote: post #116

Earlier quoted context omitted.

This kind of breaks down once you move past natural numbers, but for early learners of mathematics i agree it's useful to think about.

But other kinds of numbers can be made from that: - Differences of counting numbers (integers) - Ratios of numbers (rationals) - Limits to sequences of numbers (reals) - Solutions to polynomials made from numbers (complex) You can think of each of these as adding a layer of behavior to the basic “each number is nothing or one more than another number”.

I agree with your sequence until the last two.

The second to last, that should be Cauchy sequences.

And for the last, demonstrating that the algebraic closure of the reals is the complex numbers from first principles is much harder than describing complex numbers as pairs of reals with a multiplication rule that (a,b) * (c,d) = (ac - bd, ad + bc) and then much later proving that it is algebraically closed through complex analysis.

For those who don't know the proof, the idea is this. Liouville's theorem says that if a function is differentiable everywhere, and it is bounded, then it must be constant. Now suppose that p(z) is a polynomial. Consider the function 1/p(z). You can show that as z approaches infinity, it approaches zero. It is not constant. Therefore it must not be differentiable or not bounded. It doesn't take too much work from there to prove that it blows up somewhere, and the spot that it blows up is a point where p(z) is 0.

Apply unique factorization for polynomials (see http://sites.millersville.edu/bikenaga/abstract-algebra-2/po... for that proof) and you quickly get the fact that the complex numbers are algebraically closed.

Re: What Even Is a Number?

#145

Post by hopler »

cicero wrote: post #87
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…

This is why Aristotle said that the intellect cannot be material. The human intellect is able to receive and understand abstractions. However, whenever you try to represent an abstraction as a physical object or in a physical medium, it is no longer abstract. It is true that you can write about abstractions using physical ink on physical paper, but those are just symbols; they are not the abstraction. It is only when…

That sounds like the illusion of consciousness. Neural networks show that you don't need "dog" in order to do whatever you need to with the idea of a dog.

Re: What Even Is a Number?

#146

Post by hopler »

mzs wrote: post #141
qubex wrote: post #137

Earlier quoted context omitted.

Hate to be pedantic, but ℤ︎ is a set (not a group) and even with 0 adjoined it only forms a group under the operators of addition, subtraction, and multiplication.. If you want a set that also forms a group under division you need to ‘upgrade’ to the set ℚ︎.

ℤ︎ (…, -2, -1, 0, 1, 2, …) and + forms a group. It is also a ring with * and +. ℚ︎ with * and + forms a field.

Yes, that's what qubex said, in correcting the error in your previous comment.

Re: What Even Is a Number?

#147

Post by hopler »

ajkjk wrote: post #64
cobbzilla wrote: post #11

Earlier quoted context omitted.

I had not heard of Wheel theory, quite fascinating. But eek! From the wikipedia article: 0x ≠ 0 in the general case x − x ≠ 0 in the general case Like, you get the ability to divide by zero, but at what cost? Multiplying by zero is also not zero! Seems bonkers. Are there any practical uses for this kind of algebra?

Likely no (the peculiar distributive laws seem unusable to me). But I have come across similar rules that I think may have a real usage. It sometimes seems to me that it might make more sense to keep 'factors' attached to 0s, such that x-x = (1-1)x = 0x might be 'sound', and to keep track of 'powers' of 0s by having factors of 0 attached to 0s: (0)0 = 0^2 != 0, etc. If you keep factors like this, then you could imple…

Those "factor" zeros look like infinitesimal forms, which are well studied.

Re: What Even Is a Number?

#148

Post by bitwize »

FakeComments wrote: post #132
WilliamEdward wrote: post #116

Earlier quoted context omitted.

This kind of breaks down once you move past natural numbers, but for early learners of mathematics i agree it's useful to think about.

But other kinds of numbers can be made from that: - Differences of counting numbers (integers) - Ratios of numbers (rationals) - Limits to sequences of numbers (reals) - Solutions to polynomials made from numbers (complex) You can think of each of these as adding a layer of behavior to the basic “each number is nothing or one more than another number”.

One way to explain the real numbers to a child might be:

There are whole numbers (integers, terminology not accurate but it'll do for now), and then there are numbers called fractions (rationals) which are either whole numbers themselves, or they fall between two whole numbers, and we get them by dividing one whole number by another.

The real numbers are either fractions, or they are numbers that fall between two fractions. If it's one inch from the center of a circle to its edge, the distance around the circle is not a fraction. In inches it's about 3.14; it falls between 3 and 3 1/7, and no fraction you can possibly choose will ever get it exactly, it will be at best a little under or a little over.

You can get other non-fraction real numbers by finding square roots, but there are infinitely many of these numbers between any two fractions, not all of which are square roots. And if you try to take the square root of a negative number, you won't get a real number at all, and we call these numbers "imaginary".

Hearing stuff like this as a kid blew my mind, a bit like learning about black holes. It set me up to enjoy math throughout my life.

Re: What Even Is a Number?

#149

Post by mzs »

hopler wrote: post #146
mzs wrote: post #141

Earlier quoted context omitted.

ℤ︎ (…, -2, -1, 0, 1, 2, …) and + forms a group. It is also a ring with * and +. ℚ︎ with * and + forms a field.

Yes, that's what qubex said, in correcting the error in your previous comment.

ℤ︎ with * is not a group, there is no inverse for 0. When being pedantic it's good to be correct.

Re: What Even Is a Number?

#150

Post by [deleted] »

12elephant wrote: post #12
xamuel wrote: post #8

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…

Your castle is built on sand. What, then, is {}?

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