The ‘Objectivity’ of Mathematics

by Witty Ludwig

The first thing I need to make clear is that I am not a mathematician.  I’m not so sure it is an area in which I could ever have a great deal of proficiency.  I have, however, become increasingly interested in the philosophy and psychology of mathematics.  Given my deficiencies in this area, though, I welcome comments even more than in any other section because, being in such alien territory, I feel I am even more likely to err than in other cases.

It was during my early reading that I had a debate with a friend:  I had argued that mathematics was not in fact objective and certainly no more objective than language, whatever that would mean.  The context of this was a discussion of how, I believe, we have launched strange items into space in an attempt to communicate or demonstrate that intelligible life exists on earth, and one such item consisted of a string of mathematical formulas.  I said this was misguided since it presupposed that another life would have mathematics, would have need of mathematics, which is entirely a human convention / cultural game.  My friend didn’t necessarily disagree on this point but did proffer that I had to agree it was, at the very least, more objective than anything language could offer as a means of communication.  And so our discussion began.

His point was, let’s say that you had three walnuts in front of you: no matter what word or sound uttered, the fact that, together, they comprise the concept ‘three’ is either very simple to explain or self-evident.

This part was relatively easy to dismantle.  You only need to look so far as South America (let alone extraterrestrially), to Daniel Everett’s work with the Piraha, to see that it is a cultural tradition, habit, convention, etc. to need to group items into classifications, that is, and that cultures can exist that have no need and therefore no use for such a convention in their own life patterns.  These people could not learn how to add cardinal numbers, despite asking to be taught, because they don’t use a calculus in their day to day lives:  even their trading system worked on concepts such as ‘fewer’ and ‘more’, ‘larger’ and ‘smaller’, without the need for breaking quantities down into units.
—  To retort that they are simpler, less civilised, less intelligent etc. would not be correct.  Their faculties are the same as ours but their way of living is so far removed from how other areas of the world have developed, their way of living so different, that what is relevant to them is vastly different from what is relevant to us, thereby shaping their entire way of thinking, speaking, living.  There is no reason to believe, or not to be able to conceive, that an advanced alien form of life could not function in a similar way–  or an entirely different one from either scenario, for that matter.

Consider Daniel Tammet, for instance, who has what he describes as synaesthetic experiences as opposed to anything we would call ‘calculating’.

The second part, however, caused me to falter.  He pointed to the rim of a circular drinking glass on the table and said:  “Look, a mathematical formula can calculate the exact surface area of this glass.  I can take the length from the centre to the edge of the rim, square it, multiply it by Pi, and I will discover the surface.  This formula will work on any circle.  It is axiomatic and will apply in any and every like case.”

I knew something wasn’t quite right with what he was saying, but I couldn’t pinpoint it.  I wanted to say:  You haven’t discovered anything.  The circumference or, more accurately, that rim shape was already there to begin with.  And this, I think, is the mistake being made.  Nothing has been discovered because the proof is already present–  the space defined by the rim, the length of the rim itself, were there from the start.  All that was necessary was to create a system that maintained consistency.  What more is needed?  If, from the very beginning, the formula for a circle’s area was:  Pi(r)squared – 1 , would this cause any problems for the engineers, architects, etc.?

Wittgenstein, Remarks on the Foundations of Mathematics, I.5., c.1937-1938:

“How should we get into conflict with truth, if our footrules were made of very soft rubber instead of wood and steel?– ‘Well, we shouldn’t know how to get the correct measurement of the table.’–  You mean: we should not get, or could not be sure of getting, that measurement which we get with our rigid rulers.  So if you had measured the table with the elastic rulers and said it measured  five feet by our usual way of measuring, you would be wrong;  but if you say that it measured five feet by your way of measuring, that is correct.– ‘But surely that isn’t measuring at all!’–  It is similar to our measuring and capable, in certain circumstances, of fulfilling ‘practical purposes’. (A shop-keeper might use it to treat different customers differently.)”

What is important to recognise here is the importance culture and custom play even in something as seemingly inexorable as mathematics.

“… it can’t be said of the series of natural numbers– any more than of our language– that it is true, but:  that it is usable and, above all, it is used.”  (I.4.)