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What Even Is a Number?

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Re: What Even Is a Number?

#111

Post by dragonwriter »

stcredzero wrote: post #106
FabHK wrote: post #98

Earlier quoted context omitted.

> 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…

However, they probably have discovered numbers, pi, Pythagoras theorem, e^(i pi) + 1 = 0, etc. We can only be confident of that if they're in this universe. Also, this is only a conjecture. It's just something some of us imagine would be true. You can't elevate such an idea to the level of empirical data. So, does the fact that something is discovered (rather than invented), and can be discovered many times independe…

> 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 from the contemporary evolution of the image stem from the distant East (from the European perspective) and clearly are describing a rhinoceros. They are also very similar to accounts of the Kirin.

Re: What Even Is a Number?

#112

Post by kccqzy »

cardamomo wrote: post #97

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 won…

One of the Peano axioms is that every natural number is either zero or the successor of another. It's easy to explain to kids without jargons: a number either counts nothing at all, or it counts things that have an extra item compared to some other thing.

Re: What Even Is a Number?

#113

Post by maxbendick »

Here's a silly definition:

A number is a sequence of 1 or more digits, optionally beginning with a '-' and optionally followed by a '.' and one or more digits, where digits are the characters {'0' '1' '2' '3' '4' '5' '6' '7' '8' '9'}.

Examples: 1, 25, 3.2, -8

"1 + 2" is not a number, but its evaluation is

Re: What Even Is a Number?

#114

Post by kccqzy »

guest2143 wrote: post #2

I was surprised by no discussion of ordinality. Cardinality was something I also specifically expected. Counts of things are described here, but the greeks went to great lengths with geometry to discover numbers (and even killed people who came up with irrational answers)

Most of computer science deals with finite things. As such, it isn't necessary to introduce cardinal numbers or ordinal numbers. Natural numbers suffice. If you do, you quickly need to get into territory that requires lots of math background like well orderings, cardinal comparability hypothesis, transfinite recursion etc.

Re: What Even Is a Number?

#116

Post by WilliamEdward »

kccqzy wrote: post #112
cardamomo wrote: post #97

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 won…

One of the Peano axioms is that every natural number is either zero or the successor of another. It's easy to explain to kids without jargons: a number either counts nothing at all, or it counts things that have an extra item compared to some other thing.

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.

Re: What Even Is a Number?

#117

Post by jakeinspace »

kazinator wrote: post #109
Retra wrote: post #88

Earlier quoted context omitted.

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…

> abject poetic nonsense. No it isn't! Either it is math all the way down, or else it isn't. If it isn't, then what is at the bottom? So this is actually an existential question. Suppose we reach a final description of reality that is entirely accurate (no longer an approximation that fails to account for some structures and processes). Won't that description consist of nothing but math? Then, if it is complete, how…

My guess is that eventually, if we manage to ever reach that point of total knowledge, there are certain properties of the universe that exist for no reason; they just are. Whether that's at the level of today's fundamental particles/qft or much deeper is anyone's guess.

Example: space-time exists in n(>=4) dimensions, there are several quantum fields that exist in that space-time and evolve/interact according to the equations of qft, gravity has some mathematical description (better than the ones we have now). And that's it, there's nothing else for us to know. Of course, we can never know for certain whether we've bottomed-out our description of reality, so physicists will always have some hope.

Now, if the universe is described entirely by mathematics, bootstrapped by a few base cases (the existence of fields/particles/space-time/strings/etc), I'd be comfortable with that. I'm not sure what a truly satisfying understanding of reality would look like beyond that.

Re: What Even Is a Number?

#118

Post by FabHK »

kansface wrote: post #108
FabHK wrote: post #98

Earlier quoted context omitted.

> 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…

> They probably don't have the concept of unicorns. What would happen to your epistemology if every alien civilization has unicorns but some lack π?

> every alien civilization has unicorns

I knew it, Unicorns are real!!11!

I am not sure I can conceive of aliens that don't have pi (how can they have cylindrical rockets?).

Anyway, maybe we should let the professionals deal with this.

https://plato.stanford.edu/entries/platonism-mathematics/

https://plato.stanford.edu/entries/intuitionism/

https://plato.stanford.edu/entries/mathphil-indis/

https://plato.stanford.edu/entries/nominalism-mathematics/

https://plato.stanford.edu/entries/philosophy-mathematics/

Re: What Even Is a Number?

#119

Post by WilliamEdward »

qubex wrote: post #101
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…

I’m a mathematician and you seem a bit confused. Natural numbers ({1, 2, 3, ... }) are for counting . Giving a bunch of objects as being prime examples of that which can be counted with natural numbers is as close as one can get (short of just scribbling down the numerals and kind-of circumventing the point). Other classes of numbers (particularly the Reals) are for measuring . Again, short of writing down examples o…

I think they're just trying to say "1" is not a tangible object in the physical universe, not that it does not exist as a concept. The natural numbers are just data, information, they don't exist beyond the first dimension except as descriptors to things in further dimensions, just like lines can be built up into higher dimensions but cannot be physically created for us to observe.

Maybe you're reading too much into this.

Re: What Even Is a Number?

#120

Post by kazinator »

jakeinspace wrote: post #117
kazinator wrote: post #109

Earlier quoted context omitted.

> abject poetic nonsense. No it isn't! Either it is math all the way down, or else it isn't. If it isn't, then what is at the bottom? So this is actually an existential question. Suppose we reach a final description of reality that is entirely accurate (no longer an approximation that fails to account for some structures and processes). Won't that description consist of nothing but math? Then, if it is complete, how…

My guess is that eventually, if we manage to ever reach that point of total knowledge, there are certain properties of the universe that exist for no reason; they just are. Whether that's at the level of today's fundamental particles/qft or much deeper is anyone's guess. Example: space-time exists in n(>=4) dimensions, there are several quantum fields that exist in that space-time and evolve/interact according to the…

> there are certain properties of the universe that exist for no reason; they just are.

But we have that in math and logic: axioms!

> Of course, we can never know for certain whether we've bottomed-out our description of reality

That we do not know is a fact; but is it a fact that we can never know?

What if one fine day we do know?

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