Where Do You Want To Go To, Today?

Reading an interesting piece in this week's New Scientist about the Ultra-Finitist school of thought in mathematics -a kind of rebel, left-field, fringe movement that posits that infinity[-ies] are not at all necessary in mathematics. Not being a mathematician, I had not really encountered this philosophical tendency before, but nevertheless, I'm finding the concept interesting. I think I've talked about infinity[ies] before. Ah, yes I have, a couple of years ago. The debate would seem to hang on the difference between the concept of 'countability', ie. the basic structure of numbers themselves, and the real, physical things that we actually use numbers to 'count'. In any series of numbers; integers, real numbers, decimals, etc., it's intuitively the case that one can always increase the series by simple addition, or in the negative case, by subtraction, with no conceptual limit to the process. Fine. Simple, you might say, and suggest that one can keep counting infinitely long, bounded only by the extent of one's life, or the lives of all the 'counters' that might follow you, after your blessed release from what would have been the most Sisyphean and terminally boring life conceivable.

Now the commonly-agreed number of atoms in the known, observable universe is ten to the eightieth power, and adding in the number of their component subatomic particles raises that to to ten to the ninetieth power: inconceivably huge numbers in themselves and beyond the ken of all but the Gods themselves, should they exist. But on the number line, you can simply add or subtract another digit or decimal point to whatever limit might obtain at that particular moment for ever, without [abstract concept here] end; hence the concept of infinity[ies]. See also the post I've linked to above for the reason I include the plural in naming infinity[ies]. Suffice it to say, that a numbering system is open ended, either way, whereas our empirical knowledge of the universe is discrete and distinctly finite, and therefore countable without the need for the concept of infinity[ies]. On this point, I tend to agree with the Ultra-Finitists: pragmatically, infinity[ies] are kind of irrelevant. Unless you need them to explain some abstruse concept that lies outside [or maybe within?] the bounds of normal existence. And then you do.

Calculus, as conceived simultaneously by Newton and Leibnitz in the seventeenth century, deals with the infinite in that it is used to analyse and explain the continuous, analogue world and bring it approximately into the discrete realm of counting in order to make sense out of changes of physical state/property/position over time that can be dealt with in knowable chunks. This entails the concept of the infinite and infinitesimal division alike; a concept that lends itself to the sphere of human philosophical thought, where the imagination, being essentially boundless, can take flight in the unmeasurable infinities that we can conceive. In the real world, however, concrete measurement is required for very earthly and earthbound reasons, such as building safe bridges, railways and cars, etc. This why we have the notion of bounds within which to constrain our calculations and render them practical in discrete and measurably practicable terms.

Bounds are the limits we set within which we 'count' whatever properties/forces/motions we need to analyse, upper and lower; between which there are still infinite and infinitesimal divisions - slices, if you will - of 'thingness'. And herein lies the rub: we can restrict the overall set of stuff we want to analyse and discuss to something very finite and easily conceivable [unlike the universe or the sub-atomic world for non-physicists] and deal-able with. It doesn't mean the maths is any less difficult for most of us to understand, but the scale of the thing in question is relatable on a human level, at least: it sits within the realm of 'the real', unlike multiple infinities, angels dancing on the head of a pin, etc. 

Does this mean that the concept of infinity[ies] has no place in mathematics [or anything else come to that]? I'm not so sure; but that's just my take from a purely philosophical standpoint, and arithmetic and unbounded human imagination bears out the [human] reality of that. The known universe is just that: it is simply what we currently know from our own observations: all else is speculation, however informed. We can not know with certainty further than the current boundaries of our empirical knowledge of it. There might be more beyond, there might not: we simply don't know. And that's the thing: we can conceptually get our heads around infinity without actually being able to measure it. Just a thought: more later, methinks...