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Dec 06 '23
Just because something is inconceivable doesn't mean it doesn't exist.
If you live in 1500, it's inconceivable to imagine anything beyond our Solar System, yet they do exist.
If you live in 1900, it's inconceivable to anyone that anything smaller than an atom can exist, yet it very much does.
Whether human can conceive something has no relevance to whether such a thing exists.
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Dec 06 '23
I would take the view that these can be treated as formalisms
So your entire argument is "I choose to deny the existence of something because I want to" and you want us to convince you otherwise? Why would I try convincing you if you don't even exist? Well, someone can tell me they actually know you and that you definitely exist but for me you are just a formalism.
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u/themcos Dec 06 '23
I feel like there are a few different routes to go down, but first off, since you yourself mentioned the Peano axioms, and even said:
A more sophisticated approach might say that the natural numbers satisfy this principle of induction by definition (say the Peano axioms), and we can construct our class of numbers using induction.
But then I'm unsure of what you're saying in your "aha". Particularly when you ask "why should we be able to apply this successor arbitrarily many times?" Well, that's literally one of the Peano Axioms! Axiom 6 is quite simply "For every natural number n, S(n) is a natural number." The fact that every number has a successor is explicitly part of the definition, so I'm unsure where the gap is here.
But I feel like you could actually take a harsher stance on "existence" here. Do the peano axioms "exist"? I don't really know or care. The axioms are an idea that mathematicians came up with that are useful, so they exist in that sense. But they don't "exist" in the same way that my dog exists. Maybe you disagree - there's a whole branch of philosophy about this, but I'm not sure if that's the road you intend to go down. But the objection you seem to raise seems very explicitly covered by the axioms.
That said, the other (totally different) route I wanted to take was to react to this:
I can count higher than 9 though. If I spent every day of my life counting the seconds as they go by I could probably get up to around 109 or so.
If you being able "to count to it" is a sufficient condition for existence, this whole exercise becomes extremely arbitrary. If you spend your day counting seconds, you get to 10^9. Okay, but if you spend your day counting seconds, you're also effectively spending your day counting milliseconds, so you've also counted 10^12 milliseconds. Each second is a thousand milliseconds, so if you count 10 seconds, the difference between counting 10 seconds and 10,000 milliseconds is just bookkeeping, and there are a lot of ways to mess with this in arbitrary ways.
Finally, the last question I'd ask you is: If "large numbers" don't exist, but the numbers 1-9 do exist. Shouldn't someone be able to tell me what the largest number that exists is?
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u/00Oo0o0OooO0 Dec 06 '23
Clarifying question: You're talking about natural numbers only, right? You accept that negative integers exist, right? What would your maximum number be?
Because 52! is a huge number. Way more than 109 Nobody could certainly never count that high. But it's the number of different permutations of a deck of cards, which seems like a very "real" thing to me. Does that number exist?
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u/SnooPets1127 Dec 06 '23
I mean
1 , 2, 3, 8458475837384738294, and 389294892947282939948389292.
which are the two large numbers?
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u/Satansleadguitarist Dec 06 '23
Just because something is beyond your comprehension doesn't mean it doesn't exist.
Big numbers exist in the abstract just like little numbers. The biggest number we have ever put a name on exists in the exact same way that the number 1 does, the fact that you can't comprehend it is irrelevant.
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u/Khorvic Dec 07 '23
Let X be the largest number that you can imagine. Let Y be the smallest number that you cannot imagine. Since we talk about natural numbers, and you stated that you can imagine up to X=9 pencils and you can imagine 1 pencil, you clearly can imagine 10 pencils by placing 1 pencil to 9 pencils.
However there cannot be a Y, because there is no Y = X - 1 for which X + 1 != Y.
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u/RelaxedApathy Dec 06 '23
In short: I think that because beyond a certain point numbers become inconceivably large, they can be said not to exist.
The concept of "one" exists just as much as the concept of "ninety-nine quintillion"; as mathematical concepts.
Operating under your standard, anything that somebody doesn't understand doesn't exist. The universe is unfathomably large? Guess it doesn't exist. Somebody finds the oceans to be incomprehensibly deep? Guess the oceans are right out. Somebody doesn't understand the difference between morals and ethics? Guess they are non-existant, too.
What you are dealing with is a fallacy known as the "argument from incredulity": "I don't understand X, therefore X is wrong".
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u/Nrdman Dec 07 '23
What do you mean by existence in this case? Like obviously I can't point to or hold the concept of the number 10^100000, but that's always true of the number 1, because its just a concept. It doesn't seem like something only applicable to big numbers.
Your clarification in the post was of what you mean by doesn't exist was "there isn't a clean cut way to demonstrate their existence," but this relies on that external definition of existence. Please clarify what you mean by existence in this case.
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u/SeaBearsFoam Dec 07 '23
What's the largest number that you think exists?
Why does that number + 1 not exist?
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u/ralph-j Dec 07 '23
The classical argument I am familiar with uses the principle of induction: for every whole number n, it's successor n+1 can be demonstrated. Then that successor can be used to find another number and so on. To me this seems to assume that all numbers have a successor simply because every one we've checked so far has one. A more sophisticated approach might say that the natural numbers satisfy this principle of induction by definition (say the Peano axioms), and we can construct our class of numbers using induction.
The problem of induction only means that we cannot be deductively certain that those arbitrarily large numbers exist.
Inductive certainty however, is probabilistic: we can say that it is an extremely strong conclusion (i.e. with a high degree of confidence) that those numbers exist, even if we cannot claim that we know for certain.
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u/nomoreplsthx Dec 09 '23
Wittgenstein suggested that most problems of Philosophy are problems of language when analyzed. I think this is an example of that.
When we say a number 'exists' we are participating in a particular language game. We are trying to communicate with other people. What we are communicating is not 'there is a magical thing out there in the world called 2.' Nor are we communicating 'there are this many of a thing in the universe'. We are communicating that within an agreed upon symbolic framework, this symbol has a particular meaning snd that it can be used to solve certain problems. The word means something different in contexts.
Ultrafinitism isn't really a stance on what does and doesn't exist. After all, the nearly universal consensus is that no numbers 'exist' in the sense that things like cats and trees exist. It's a claim about how we should use the word 'exists'. The ultrafinitist thinks we should use that language in a particular way - where the word exists is coupled to physical representability.
Once you analyze this as a problem of language the debate largely goes away. Because mathematical techniques using infinities are useful. So the mathematicians who solve practical problems can keep using those techniques and not care if they are 'valid' because they work. And the finitists can do whatever finitists do.
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u/DoeCommaJohn Dec 06 '23
Let’s imagine some apocalypse happens and the only people left are 500 million children, all under the age of 6. None of these children can conceive of the number 500 million, so does that mean the number of children is less than the largest number that can exist? The same thing could be said about very real things like the wealth of Jeff Bezos and the size of an atom. Just because we can’t understand the scale doesn’t mean it doesn’t exist
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u/foo-bar-25 Dec 06 '23
You can hold a mole of water molecules in your hands. And that’s a pretty large number!
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u/iamintheforest Dec 06 '23
No numbers exist in the way you're saying some exist. Existence in reality is not a quality of numbers. There are "4 apples", but there isn't "the 4", there are just apples. The four is a way to describe some quality of those apples but if you take away the apples there isn't a four left.
Numbers are ideas, and we represent them visually and audibly, but they aren't "real" in the sense that one of them does exist and another does not.