## Doing Maths Without Numbers

Define ‘one’. Go on, try.

Well, ‘one’ is what you have when there’s only one of something. You have one duck. There – one. But that’s circular. You still need to define ‘one’ for you to have one duck.

All right, it’s one more than zero. ‘One’ more? Circular again, and what’s this ‘zero’ you’re talking about now?

Half of two? Okay, fine, but what’s two? Well, that’s twice one, isn’t it? Circular again. People trust mathematics with their lives, and yet all it seems to rest on is circular arguments. Can’t we do better than this?

Let’s start at the very beginning, with something really fundamental: “things are what they are.” A duck is a duck. A submarine is a submarine. “It is what it is” is a property shared by all things. Let’s call this the property of *identity*, of *being identical with itself –* the property of being what it is.

Everything is identical to itself. If you grouped together everything that is identical to itself, the set would have *everything *in it. And the set of everything *not* identical to itself would be empty.

We have defined an *empty set* – a group with no members. The set of four-sided triangles is empty, because nothing is both a triangle and four-sided. So is the set of things that are not identical to themselves, because nothing is.

Now we’re used to the idea of sets, consider this: what does the set of three ducks have in common with the set of three triangles? Call it the *cardinality* of the set – a measurement of how much is in it.

Imagine the set of all empty sets. What do all these empty sets have in common? Let’s call it *zero* – the cardinality of an empty set. We have now defined zero. If something is the same as zero, then it has the same cardinality as the set of all things not identical to themselves.

Now that we know what zero is, how many things are identical to zero? There’s just one thing that’s identical to zero: zero. If you filled a set with things identical to zero, there would only be one thing in it. The cardinality of the set of all things that are identical to zero, is one.

And the cardinality of the set of all things that are identical to ‘one’ or ‘zero’ is two. And the cardinality of the set of all things that are identical to ‘zero’ or ‘one’ or ‘two’ is three, and so on and so on and so on.

To break the circularity of mathematics, Bertrand Russell defined ‘I have two hands’ as:

“There is such an *a* such that there is a *b* such that *a* and *b* are not identical and whatever *x* may be, ‘*x* is a hand of mine’ is true when, and only when, *x* is *a* or *x* is *b*.”

And this is how you do maths without numbers.

REFERENCES

The BR quote was from *A History of Western Philosophy* (1945); also see *My Philosophical Development* (1959).

The layout of the above argument was from a YouTube video on mathematical logic, but I can’t remember where it is; I’ll update this if I find it.

Disclaimer: I know noticeably less about philosophy than you. And I don’t overestimate you in that.

I got halfway through it by the time I thought of: “‘One’ is the condition by which something is in some set containing nothing else”.

It helps to recognise that when we say ‘one’, it is an abstract concept which stems from a reality in which the ‘thing’ has context. (One duck *in the room*).

Also, I find cardinality circular. “Three is the property defining the set ‘things in threes'”. This is the process that makes me find the rejection of axioms problematic.

On the whole, though, very well described ideas.

DTysenAugust 14, 2011 at 2:46 am

Yeah, cardinality is a tricky one to talk about and keep the word count down, and the way I phrased it does make it seem circular. Here’s a more thorough definition (adapted from the Wikipedia page on cardinality):

Consider two sets, X and Y.

X contains (a b c d e)

Y contains (f g h i j)

Now perform a function in which you take all the members of X and pair them with members of Y.

a-f

b-g

c-h

d-i

e-j

If you remove all the members of X that have been successfully paired with a partner with Y, you find that X is now an empty set. You can map the contents of X onto Y, meaning that Y has at least as many members as X. Call this property A.

And you also find that if you perform the same operation on Y, pairing its members with X, this leaves Y an empty set, meaning X has at least as many members as Y. Call this property B.

If a pair of sets have both property A and property B, then they have the same cardinality. The sets have the same cardinality if and only if each one can be mapped onto the other without leaving a remainder.

http://en.wikipedia.org/wiki/Cardinality

http://en.wikipedia.org/wiki/Bijection

The S IAugust 14, 2011 at 10:18 am

“what’s this ‘zero’ you’re talking about now?”

i always thought of zero as “the infinitely small” universe as in the other side or direction of “the infinitely large” universe, but then “thought” like time and space could also be a construct of the mind.

philosophically speaking i also wonder 😉 what the mind of the dolphin once they evolve to our current level in tech and science will think about this zero they invented for heir math… grin, thanks for all the fish

mad profOctober 6, 2012 at 9:34 pm