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What Even Is a Number?

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Re: What Even Is a Number?

#91

Post by qubex »

mjburgess wrote: post #15
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…

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…

This reminds me of my Italian grammar teacher telling me that nouns can be broken into two categories: ‘concrete’ and ‘abstract’ (for example, ‘chair’ and ‘love’ respectively). Then came a totally mind bending digression for my eight-year-old mind when she mumbled “well actually all nouns are abstract because there’s no such thing as a chair divorced from reality, there’s nobly instantiations of these abstract concepts we consider close enough to be concrete”. That totally blew my mind. Together with reading The Computational Beauty Of Nature” when it first came out nine years ago those were the two key experiences that drove me towards choosing to get a degree in mathematics.

Re: What Even Is a Number?

#92

Post by tim333 »

toddwprice wrote: post #21

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 w…

It may be more than a language to describe the world. Take Euler's identity, e^(i*pi)=-1. One of the most striking equations in maths and it doesn't really describe anything physical in particular and true regardless of the nature of the world.

Re: What Even Is a Number?

#93

Post by tim333 »

As a follow on people may enjoy Feynman's lecture on Algebra http://www.feynmanlectures.caltech.edu/I_22.html and some audio clips http://www.feynman.com/the-animated-feynman-lectures/

>To discuss this subject we start in the middle. We suppose that we already know what integers are, what zero is, and what it means to increase a number by one unit. You may say, “That is not in the middle!” But it is the middle from a mathematical standpoint, because we could go even further back and describe the theory of sets in order to derive some of these properties of integers. But we are not going in that direction, the direction of mathematical philosophy and mathematical logic, but rather in the other direction, from the assumption that we know what integers are and we know how to count. ...

Re: What Even Is a Number?

#94

Post by kevin_thibedeau »

toddwprice wrote: post #21

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 w…

Sine waves do exist. They are the 2D projection of a circular spiral, which is fundamental to the properties of the physical world. This is rarely explained fully when these concepts are introduced in trigonometry (which should more properly be called "circleometry") because of the higher level maths required before introducing imaginary exponentials.

The computer your are using would not work without sine waves. The nanometer scale chips couldn't be manufactured without them. The high speed signals transmitted over its wiring would not reach their destination without knowing how to manipulate them.

A deeper question is what to make of "uncountable" transcendental numbers.

Re: What Even Is a Number?

#95

Post by mzs »

tim333 wrote: post #92
toddwprice wrote: post #21

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 w…

It may be more than a language to describe the world. Take Euler's identity, e^(i*pi)=-1. One of the most striking equations in maths and it doesn't really describe anything physical in particular and true regardless of the nature of the world.

I can visualize it as triangles.

https://www.math.toronto.edu/mathnet/questionCorner/epii.htm...

Re: What Even Is a Number?

#96

Post by tim333 »

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…

Numbers exist in a different way to Sherlock Holmes who of course exists as a fictional character. One of the things people look for when searching radio signals for alien life is prime numbers as intelligent species would discover those but they are not produced by mechanical processes like quasars. How could we and they discover them if they don't exist at least in some form?

Re: What Even Is a Number?

#97

Post by cardamomo »

I was once posed a related question by a three-year-old: "How do you make a number?"

It was a simply worded but delightfully insightful wondering. I did my best in the moment and answered, "One way we can make numbers is by counting." I've continued to mull this question over since then. I'm drawn to this child's conception of numbers as something we make. It makes sense to me as a teacher of young children and a wonderer about math myself, yet it doesn't really square with what we've been told about numbers in most educational settings.

Re: What Even Is a Number?

#98

Post by FabHK »

stcredzero wrote: post #35
kazinator wrote: post #25

Earlier quoted context omitted.

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.

I believe that in fact the numbers do exist as physical objects exist. Just in their own universe. 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…

> 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 than invented), and can be discovered many times independently in identical fashion, confer it some sort of elevated ontological status?

Re: What Even Is a Number?

#99

Post by incompatible »

Retra wrote: post #88
kazinator wrote: post #25

Earlier quoted context omitted.

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.

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…

Yeah, mathematical objects like numbers are concepts in our brains, or imaginary objects. Numbers in particular are imaginary objects used for counting. You can set up an isomorphism between these numbers and real-world objects, e.g., to count them.

Re: What Even Is a Number?

#100

Post by galaxyLogic »

Number is an abstraction. What is an abstraction? It is a definition for a set of things by stating what is common to all things considered to be in that set.

"1" is the set of all things which have exactly one element. It is an abstraction for all such things.

What does it mean to "have one element"? I think that's a more difficult question. Maybe it should be an axiom?

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