Reason and Belief in Mathematics and Neomathematics

It is interesting to consider the foundations of mathematical belief.

In some cases it is primarily Pavlovian. basic arithmetic is induced in people by rote memory pumped in through early school reward and punishment. People believe 2+2=4 because they have been told to do so. Even in those of us who may have developed their own mathematics in preschool years, if they are given over to society's gentle ways, they develop deep archaeological strata of socialization, and will tend to believe, if not entirely, the social definitions of mathematics.

We might form a model of 2+2=4 by placing two sets of two pennies each beside one another. Then we see four pennies through a fairly direct perceptual experience. But how much of that fourness is reality and how much socialization? Certainly the verbal and mathematical names 'four' and '4' are social. is the thing-number itself social?

Exercise: Place four pennies on a table in the dark. Try to convince yourself that there are five, or three, pennies present. Notice your strategies of disbelief.

What is the nature of slightly more complex statements? For example, 4+26=30. This is not something which is necessarily obvious at first glance, in terms of randomly arranged objects. It takes counting or mathematics to ascertain.

What makes us think that 4+26=30? Is it a matter of the psychology of belief or is it mathematical truth? Suppose we had 26 pennies in a pile and added four more, and then counted them and found there to be 31 pennies. it could appear that either we made a mistake, or that another penny came into existence (or perhaps slipped onto the table by a predigitator), or that in this case, 26+4=31. The first possibilities give no problem here, but what if it is the case that, while usually 26+4=30, sometimes 26+4=31? this answer offers no empirical difficulties; in some ways it is more logical than the mistake hypothesis. Suppose that instead of 30, we have 30,000 pennies. How do we know that we have 30,000 pennies? Someone, either a machine or a person, perhaps ourselves, counted them and told us so. How likely are they to be correct? Not too close to certainty, I imagine. if we count the pennies several times and always get the same answer, then that answer seems likely to be the case. but then just try having a few other people count the pennies! (And why would they want to do that?)

So we see that a non-static mathematics, at least for elementary arithmetic, is empirically justifiable. Is there then any fundamental mathematical reason for not accepting that sometimes 26+4=31? You could set up a Peano system and construct 26+4=30. Does that make it universally so? It only puts that fundamental question which I am trying to approach one step further back in logical regression. Why should you believe any step of the construction?

We believe such things either because we are told to, or because we have some deeply grounded intuitive perception of certain very direct mathematical or logical truths. More generally and more accurately, we believe certain mathematical facts (I almost typed 'fables', FJS, 2000) because they are entwined in a richly green, brown and complex forest of interwoven meanings and personal, social and historical events and personal experiences.

Can you see what a wild, primitive jungle mathematics as it now is, is? Axiomatics just adds to the jungle, mostly parasitical creepers on the true hidden trees of unfound deep theorems. And it is barely twilight! Compared to mathematics, neomathematics is a sunlit clearing, although at first one might be blinded by the brilliance.

Neomathematics radiates much more directly from actual mathematical experience of the world and of consciousness than does formal mathematics. In its partial vagueness, it is thus precisely applicable to real experience and allows flexibility in attuning mathematics to nature and thought.

Consider a non-static analysis of seminfinity. We presently characterize seminfinity as a number large enough so that, by greatest efforts, a personal or machine might be just able to count toward it. There will always be some uncertainty associated with this large counting and, in general, the larger the counting the larger will be the uncertainty. We can imagine a seminfinity large enough that any given finite number may be taken as the uncertainty. Thus seminfinity, N, is static in the large, but is locally variable: N = N +/- n , for any finite n. In actual human one by one counting, seminfinity may become rather small, perhaps on the order of a thousand.

In these contexts we can see neomathematics as a conceptual reservoir from which mathematical concepts may be drawn and delimited as needed.

Joe Staley

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