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How inevitable is the concept of numbers?

writings.stephenwolfram.com

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Re: How inevitable is the concept of numbers?

#41

Post by falcor84 »

Ekaros wrote: post #34
paganel wrote: post #2

I can understand how natural numbers can be “constructed” (for lack of a better word) as a byproduct of counting, what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero. Maybe related, while I can also partially understand multiplication (syntactic sugar for adding) I could never understand multiplication by zero, meaning how come when you mu…

Start with then fingers. Then lose two. You have now 8 fingers. What is the number of lost fingers if not negative number.

What do you mean? The number of "lost" fingers is 2.

Re: How inevitable is the concept of numbers?

#42

Post by cabalamat »

paganel wrote: post #2

I can understand how natural numbers can be “constructed” (for lack of a better word) as a byproduct of counting, what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero. Maybe related, while I can also partially understand multiplication (syntactic sugar for adding) I could never understand multiplication by zero, meaning how come when you mu…

> I can understand how natural numbers can be “constructed” (for lack of a better word) as a byproduct of counting

So you start off with zero, and the successor to zero, and the successor to that number, etc, and they're the counting numbers...

> what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero.

...and there are all sorts of numbers that aren't counting numbers, but can be manipulated by the same rules of arithmetic. E.g.:

    h*2=1 -- fractions

    n+1=0 -- negative numbers

    r*r=2 -- irrational numbers

    i*i+1=0 -- imaginary numbers

Re: How inevitable is the concept of numbers?

#43

Post by falcor84 »

paganel wrote: post #2

I can understand how natural numbers can be “constructed” (for lack of a better word) as a byproduct of counting, what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero. Maybe related, while I can also partially understand multiplication (syntactic sugar for adding) I could never understand multiplication by zero, meaning how come when you mu…

As a kid, I recall that the thing that really made negative numbers click for me was underground floors, particularly when pressing on elevator buttons.

Re: How inevitable is the concept of numbers?

#44

Post by [deleted] »

contravariant wrote: post #28
SquibblesRedux wrote: post #4

While the post covers quite a bit of ground, it feels (to me) like it conflates knowledge representation, language, biological systems (i.e., the messiness of implementation), computability, and realism. Regarding numbers in particular, there are a practically uncountably infinite number of mathematical truths that apply equally to numbers or to other abstract (non-numerical) mathematical ideas. I would rather see de…

Because of the limitation of language there's only a countable number of mathematical truths that can be proven or written down. So for all practical purposes there's countably many.

[deleted]

Re: How inevitable is the concept of numbers?

#45

Post by joshuaissac »

paganel wrote: post #2

I can understand how natural numbers can be “constructed” (for lack of a better word) as a byproduct of counting, what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero. Maybe related, while I can also partially understand multiplication (syntactic sugar for adding) I could never understand multiplication by zero, meaning how come when you mu…

We can informally derive other types of number by extending operations on our existing set of natural numbers, seeking a "closure" for that operation, and seeing if we get consistent results.

So extend the concept of differences between natural numbers, by subtracting a large number from a smaller number and defining the result as belonging to a new class of number. Similarly, get fractions by defining non-integer ratios between integers, real numbers by defining non-fractional limits of infinite sums of fractions, imaginary numbers of defining non-real roots of polynomials with real co-efficients, and so on.

Each time, we have an operation that works for some subset of our numbers, so we look at what does not work, and see if we can make it work anyway by defining the result of such an operation as a new type of number. And then we repeat with a different operation.

But there are things we cannot consistently "extend", like division by zero, or 0^0, so we leave just those out.

Re: How inevitable is the concept of numbers?

#47

Post by Aardwolf »

contravariant wrote: post #28
SquibblesRedux wrote: post #4

While the post covers quite a bit of ground, it feels (to me) like it conflates knowledge representation, language, biological systems (i.e., the messiness of implementation), computability, and realism. Regarding numbers in particular, there are a practically uncountably infinite number of mathematical truths that apply equally to numbers or to other abstract (non-numerical) mathematical ideas. I would rather see de…

Because of the limitation of language there's only a countable number of mathematical truths that can be proven or written down. So for all practical purposes there's countably many.

Could you prove for every real value between 0 and 1 that it's greater than or equal to 0? That's an uncountable amount of proofs

Re: How inevitable is the concept of numbers?

#49

Post by kevinventullo »

Aardwolf wrote: post #47
contravariant wrote: post #28

Earlier quoted context omitted.

Because of the limitation of language there's only a countable number of mathematical truths that can be proven or written down. So for all practical purposes there's countably many.

Could you prove for every real value between 0 and 1 that it's greater than or equal to 0? That's an uncountable amount of proofs

No, because proofs have to consist of a finite number of words. Thus there are only countably many proofs of anything. In particular, there are only countably many reals between 0 and 1 which can be expressed in a finite number of words.
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