Most people believe that certain things or ideas are just so, no evidence needed. Some, who should know better, take this common sense idea to ridiculous extremes and claim that this kind of intuitive knowledge is derived from a Godlike source, and by adherence to strict rules of logic, universal truths will show themselves. These geniuses, no irony intended, suggest that, since this knowledge is absolutely certain, it is superior to that derived from experience and science. Certain philosophers and mathematicians seem to be especially enthusiastic about this mode of thinking.

I have some expertise in biology, much less in math and philosophy. Nevertheless I will singlehandedly confront this myth, and expose it for what it is: an atavistic remnant from prehistory when gods, devils and evil spirits ruled the lives of innocent men and women. I trust we are a little less innocent today.

A PRIORI KNOWLEDGE of the TRUTH? Much of what passes for knowledge is inspired guessing. I would submit that humans do not even have a priori beliefs. Human beings do have an abundance of innate abilities but it is not clear what that has to do with knowledge. Newborn babies, tiny hominids that do not exhibit consciousness, can tell the difference between sweet and sour. That is not surprising at all. Some months later while still in a preconscious state of development, infants will react uncomfortably to situations that appear unfair: an innate sense of justice appears to be wired into our nervous system. This is also a far cry from a recognition of truth and the toddlers would be unable to articulate their judgement on the matter.

Some extremists take this to mean that we have access to TRUTH about the universe independent of our sense experiences. Exhibit number one for their case are the surprising truth claims derived from mathematical axioms. It is indeed disconcerting to us real world people that these ideas are widely accepted. “Yes, math is deployed as a tool in science and in all sorts of other applications, but there are huge swaths of mathematical territory that neither describe anything in the world nor are pursued by mathematicians for any practical reason at all.” – Massimo Pigliucci, Scientia Salon: The return of radical empiricism.

The first question to ask about math axioms is how certain are they, and are there any conceivable inaccuracies? The slightest smidgen of a whisper of an imprecision or inaccuracy would make the extreme extrapolations of math completely unreliable. In biology we are paradoxically advised to “never say never, and never say always”. We also know that we cannot extrapolate beyond the limits of our system of observation – what holds within the range of our observations frequently is compromised by unexpected behavior outside that range. Math is not concerned with these caveats. Their ‘truths’ are supposed to be universal and unchanging, but we should be skeptical of anything so pure and so powerful, yet so mysterious. There may be hidden distortions and imprecisions in some axioms occasioned by unconscious anthropomorphisms and anthropocentrisms.

A more serious problem with axioms and mathematics, for me, is that they are completely abstract. In the real world I may have one orange (A) and you may have one orange (B), so we are equally supplied with delicious fruit (A=B). This is true in human and real world terms where we have no choice but to be satisfied with approximations; however, no two biological or physical structures are identical and in pragmatic terms we never achieve absolute equality (A never=B). Absolute numerical equality occurs only, and by definition, in an abstract mathematical universe where entities have no substance. The mind can travel from zero to infinity in a blink of the eye, the only effort required is to understand the language and rules of the game. The activity is pure reason and is completely and utterly different from the real world. What significance then could there be to an abstract mathematical ‘truth’? It is a pure logical truth and is unlikely to describe anything in the particular universe that is inhabited by human beings. I don’t know about other universes or n-dimensional space.

Is a mathematical truth a truth in human terms? The answer must clearly be no. Human truth is extremely hard to come by, it is fragile and appears to change. Pure mathematical truths, sui generis, are logical abstractions and provide satisfaction to an extremely small cohort of practitioners.

So, it is a good question as to why do some mathematicians and philosophers get so enamored of these mystical and strange ideas emanating from their abstract ruminations? My guess is that many people are emotionally attracted to religious, mystical and/or occult thinking, whatever that means. That is a perfectly fine thing to do, it should just be understood for what it is.

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