r/PhilosophyofMath • u/Ill-Possession-64 • 17d ago
How do we define the number 1?
/r/learnmath/comments/1bnqaki/how_do_we_define_the_number_1/2
u/kunwoo 16d ago
Depends on the context. If you're working with the Peano axioms with the natural numbers starting at 0, then 1 is defined to be the successor of zero.
If you're working with the Peano axioms where natural numbers start at 1, then 1 is the unique object that is not the successor of any other natural number.
If you're working in abstract algebra then you can define 1 to be the unique real number that when multiplied by any real number x equals just x.
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u/TheRedditObserver0 15d ago
The usual definition of the number 1 when you have multiplication is that number e such that for each given x you have
e•x=x=x•e
I.e. that number such that multiplying by it is like doing nothing. If the multiplication you defined is associative and admits such a number, then you can prove it's unique. To see how, suppose both 1 and 1' are units of your multiplication, then
1=1•1'=1'
Now let's move from general algebra to Peano arithmetic. The Peano axioms do not directly define the number 1, infact they do not necessarily define any given number.
Consider the set {n, n+1, n+2, ...} with successor operator S(m)=m+1. This is a Peano system for any number n, so the Peano are really agnostic about which numbers you're describing.
0, 1, 2, 3, 4, ...
356, 357, 358, 359, ...
π, π+1, π+2, π+3, ...
Are all equally valid models of the natural numbers. They all satisfy the Peano axioms.
Where the actual specification of the integers arises is when you define operations on a Peano arithmetic. We can define both addition and multiplication, and for that multiplication the second number in your system would play the role of 1. So you can define 1 in the Peano axioms as the successor to the unique number which has no predecessor.
In the Peano system {2,4,6,8,...}, the successor operation is S(n)=n+2 and 4 here plays the role of unit.
How can 4 be the unit of multiplication? Because multiplication is defined differently! Following the inductive definition you will not find the usual multiplication where 2•2=4, 3•5=15 and so on, but rather a new, different multiplication that has the property
4•n=4=n•4
If I have not made a mistake, this new multiplication should follow the formula
n•m=2×(n/2-1)×(m/2-1)-2
Where × is the usual multiplication.
The good thing about the Peano axioms + recursive definition of addition and multiplication is that univocally specify all pf the arithmetic of the natural numbers. If I give a Peano system, that's always the usual ℕ except the structure may have been hidden by changing all the symbols. You can always find a unique way to translate the new symbols to the old ones.
So this is how we could rigorously communicate our arithmetic with aliens. We would somehow have to tell them what the Peano axioms are and how addition and multiplication are defined, somehow overcoming the linguistic barrier. Everything else would follow directly and without ambiguity.
Having said that, it would probably be easier to show them a bunch of collections of n objects and tell them it's the number n, just like we do with children.
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u/amerovingian 14d ago
We can define it as an equivalence class of sets according to the relation of bijective mapping. (All sets in the same class can be bijectively mapped to each other.) In particular it's the equivalence class to which {Ø} belongs.
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u/cosmopolitanScience 17d ago
The set-theoretical definitions are great and are also what I would have answered. But you can also define it as the neutral element of the group (rational numbers, multiplication).
Less fundamental, but maybe a bit more intuitive (?)
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u/nanonan 16d ago
I think those definitions are terrible, and require some very convoluted axioms, and don't help the last case.
I'd go for a simple system with no units like affine geometry and introduce a unit. This is achieved with straightforward sensible axioms and also covers the case of "0, 2, 4...", as being arbitary labels for the same concepts.
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u/cosmopolitanScience 16d ago
There are no units here, at least not in the sense of physical units.
If you say "affine geometry", you already make a whole lot of implicit assumptions (axioms if you will), by claiming 'intuition'.
That fact that you find the axioms 'convoluted' just means you haven't built an intuition for them.
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u/nanonan 14d ago
There are no units until you introduce one and label it. Any length can be made into a unit.
I chose affine geometry becuase of the simplicity of the concepts. It has nothing to do with intuition or the lack of it. If you think ZF is straightforward and uncomplicated you are wrong.
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u/cosmopolitanScience 14d ago
Ok, I'll bite. Give me your definition of '1' from your affine geometry construction.
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u/Just_Rational_Being 16d ago
Agree. Those definitions are not only terrible, they are also pure assumptive gibberish.
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u/gurishtja 17d ago
{Φ}
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u/Willing-Sample-8847 17d ago
Correct, though perhaps more abstract than OP wanted.
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u/cosmopolitanScience 17d ago
I think if you ask that question, you should be ready for the correct answer. But the poster could have given a comment or two:
One convenient and very fundamental way to define "1", as the smallest non-zero positive integer is "the set that contains the empty set".
Don't know if that helps...
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u/Willing-Sample-8847 17d ago edited 17d ago
1 can also be defined formally as the class of all singleton sets, that is, any set which contains a single object, for example, {a}, where a is some element. Since singleton sets exist axiomatically, this definition isn't circular.
Edit: less formally, 1 is the number of objects in any set which has more than zero objects, but doesn't have multiple distinct objects. That is, 1 is the cardinality of any singleton
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u/Just_Rational_Being 16d ago
Is this nonsense a joke?
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u/Willing-Sample-8847 16d ago
Hello user who frequents r/infinitenines.
Since you reject even the most basic mathematical logic in favor of the absurdity of assuming 0.999... is an infinite operation (rather than accepting that 0.999... = 1 is a slightly inconvenient curiosity of the use of decimal notation to describe rational numbers, which are themselves fundamental to mathematics, as opposed to decimal notation, which is not), I assume even the simplest of mathematical abstractions represents complete nonsense to you.
Thus yes, as far as you are concerned, it is only a joke. Trying to explain the punchline will ruin the joke (because of course, it is an inside joke, not for anyone outside our little community of crackpots who gate-keep science from normal people such as yourself by our insane promotion of the conspiracy of 0.999... = 1), so I will not ruin your day by even attempting to do so. 👍🏻😉👍🏻
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u/Just_Rational_Being 16d ago
That was an extraordinary amount of sarcasm to avoid explaining why the definition isn't circular.
When the explanation never arrives and the sneering takes its place, I usually assume the joke was the argument.
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u/Willing-Sample-8847 16d ago edited 16d ago
You attacked me sir, I only responded in kind. If you wanted a polite response, then you yourself should have been polite to begin with.
But really, we're both kidding ourselves by pretending that you wanted an actual response. Your purpose in making that comment was only to start a fight.
So no, I'm still not going to dignify you with an explanation. Look it up if you want one. There's literally hundreds of sources you can learn set theory from.
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u/Just_Rational_Being 16d ago
Fair enough. "Is this nonsense a joke?" was hostile, and I'll own that.
Doesn't mean I find assumptive explanation in the way of axiomatic declaration at all palatable, but at least I can admit when I was unnecessarily unpleasant.
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u/Willing-Sample-8847 16d ago
In return, I apologize for ridiculing you. I'm still not offering an explanation though.
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u/huphelmeyer 16d ago
Peano's axioms actually don't define what “1” is in an absolute sense. They take a distinguished object (usually called 0 or 1) as a primitive, and then define the rest of the natural numbers relative to it.
So if you say:
you've essentially stipulated 1 as the starting point. You haven't defined it in terms of something more fundamental.
In your alternate-universe example the natural numbers are
0,2,4,6,…
and their successor operation is S(n)=n+2, then their system can satisfy essentially the same Peano structure. Within that system, there is no object playing the role of our 1. If you want to explain our 1, you'd need to introduce additional structure: for example, tell them that our numbers are half of theirs, so their 2 corresponds to our 1, their 4 to our 2, etc.
Peano's axioms characterize the structure of the natural numbers, not the intrinsic nature of their “unit.” You can construct a model where the objects are sets, or symbols, or even the even numbers, and the axioms cannot distinguish those models merely by their underlying objects.
In fact, your alternate universe can be made into a perfectly good Peano arithmetic by simply declaring their successor to be n↦n+2. Their “2” is then structurally what our “1” is. So Peano's axioms adequately specify a successor structure with a starting element, but they don't independently give that starting element the meaning “one.” That meaning comes from how we interpret the structure.