Plurality of arithmetics? Or absoluteness of arithmetical truths? (i. e. Are all mathematical truths actually conditional?)

Question

The following assertion is attributed to Russell ( as a quote from Mysticism and logic) :

Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. [...] Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

I think I understand in what sense this assertion is true of geometry.

But is it actually true of arithmetics?

Of course, one could argue that arithmetical truths qualify as truths in an arithmetical system.

In that sense, I should not say categorically that " 2+2 =4 " but that

" IF [ some definitions and properties of addition] THEN 2+2=4".

But, as far as I know, all these defintions and properties that play the role of " hypotheses" are not considered as hypothetical by mathematicians.

I've never heard about any " alternative arithmetical systems", or " non-classical arithmetics". Does such a thing exist?

So, cannot one say that "2+2=4" is true, period? ( I mean , categorically, not hypothetically)?

Or, at least, definitions of numbers being admitted, cannot one say that " 2+2 = 4 " is categorically true?

Answer 1

All of math comes with conditions. Here's a list of different and related arithmetics:

• Presburger arithmetic
• Peano arithmetic
• Robinson arithmetic
• Second-order arithmetic
• Modular arithmetic ($2+2\equiv 1 \bmod 3$)

This list is far from complete. All either generalize other arithmetic, or have different conditions. Some not included ( probably subsets of more areas of math), would be p-adic arithmetic, matrix arithmetic, vector arithmetic, polynomial arithmetic, etc.

• All arithmetic rules you might learn in school, need not apply generally in anything less than a specific Field for example.

• All we prove with proofs, is consistency with a given framework. Unless it works in every framework it could be put in. See Gödel

Answer 2

Suppose that after a short rain, you observe drops on your window. There is $\mathbf 1$ drop here and $\mathbf 1$ drop there. Now they flow together and form ... $\mathbf 1$ slightly bigger drop. Did we just show that $1+1=1$?

• Does each of the original drops by itself and the resulting combined drop represent the notion of "$1$"?
• Does "flowing together" represent addition? If not, why not? If only because this does not lead to $1+1=2$, the argument may be circular
• Is "$=$" correctly represented when we actually compare different things, namely the srops before vs. the drops after the flowing together?

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Two shepherds have big herds of sheep, say one in A-town has $2327$ sheep and the other in B-town has $1729$ sheep. They agree to combine their herds and to this end drive them to C-town in the middle. Is it clear that they will count $4056$ sheep?

• To begin with, is the idea of joining herds of sheep what you would call "$+$"? Probably yes - but in our case we were fortunate that the original herds were disjoint ...
• As above, it may be doubtful that "$=$" is the right thing to use for this process when we have yesterdays counts from A-town and B-town (i.e., before they drive the herds together) on one side and todays count from C-town on the other. In fact, special relativity will tell you that the result counting things that are not in the same place may depend on the oberver's motion ...
• At any rate, it is well possible that some lambs were born or some old sheep died between the two counts, hence we cannot be full confident that the result will be $4056$ ...

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And sometimes we explicitly note that "the whole is more than the sum of its parts".