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.

You are possibly confused between “a priori” and “innate”. A belief can be a priori, even though it is not innate.

This is why many people consider mathematical truths to be a priori. They do not require empirical evidence. Perhaps you would need to see a proof, but that’s not empirical evidence (it is, perhaps, logical evidence).

I think you would do better to say that it is not an empirical truth. Many humans do consider mathematical truths to be true, so that seem to fit as “a truth in human terms”.

Hi Neil,

I have become very skeptical of, even antipathetic to, the inflated claims of metaphysicists and theoretical mathematicians. Every now and then they claim to see a universal truth or understand the mind of God. When challenged, they retreat under the protective shell of their professional jargon – they start speaking in tongues, as it were. I say it is time for them to start communicating in ordinary language. I would enjoy the public defrocking!

“A belief can be a priori, even though it is not innate.”

Where would an idea come from if it were not from experience via all our senses? Human beings have unlimited imaginations and will see a cosmic unity of consciousness, panpsychism or god. These abstract mystifications are entertaining but become extremely dangerous when they infiltrate the minds of charismatic egomaniacs.

I would say it is much simpler and much more likely that if an idea is not of innate origin, it is a posteriori. The empirical ‘truth’ is that all ideas are the result of innate processes that have been adapted to deal with our internal and external environment. In the case of some math and metaphysical ideas, the connections to these bodily processes have become obscure leading to their misinterpretation as a priori.

Do you know of any a priori knowledge derived from pure math or metaphysics that has changed the empirical universe? I would be curious.

A definition is true by definition. So its truth is known a priori. It may well be that the definition is inspired by experience. However, experience my inspire the definition but it does not logically dictate or imply the definition. Once we accept a definition, its truth is obvious a priori (without empirical evidence).

I’m not suggesting that you are confused about where beliefs originate. Rather, that you are not sufficiently familiar with the way that “a priori” is used in the literature.

Liam Uber: “Most people believe that certain things or ideas are just so, no evidence needed.”

A big, big problem about this statement. What is ‘evidence’ for you? Every (every, …, every,… ) argument always shows some ‘evidences’ while you might see them as hogwash. It is your bad.

Liam Uber: “A PRIORI KNOWLEDGE of the TRUTH?”

‘A priori’ is an outdated old school concept. When here is a question (problem) and I give you the answer, the key is about the validity of my answer. How I get my answer could be very simple or very complicated, and the process cannot be encompassed by the stupid old concepts of a priori or else. My post (http://scientiasalon.wordpress.com/2014/08/28/the-return-of-radical-empiricism/comment-page-1/#comment-6855 ) can be one example of this.

Liam Uber: “I have some expertise in biology, much less in math and philosophy.”

This is the major problem. Without truly understanding math, there is no way for you to know the difference.

Liam Uber: “The first question to ask about math axioms is how certain are they, and are there any conceivable inaccuracies?”

Absolutely not! There is no ‘right or wrong’ issue in math-axiom but has ‘good or bad’ distinction. A dad math-axiom system is often meaningless and having many contradictions, but it is not a proof that that axiom is ‘wrong’. My post (http://scientiasalon.wordpress.com/2014/08/28/the-return-of-radical-empiricism/comment-page-2/#comment-7001 ) gives a more detailed explanation on this.

Thanks Neil Rickert, Tienzen (Jeh-Tween) Gong,

I am learning and your comments help. As an eclecticist interested in the human condition, I keep an eye out for possible sources of vital information. I have become less interested in the details of philosophy and more interested in its process. I am really studying the means of acquiring and sharing information, and how this relates to decision making and behavior. Studying biology seems to provide much more insight than some of the other disciplines. Psychology is interesting, so is neuroscience. Metaphysics has been a disappointment so far, mainly because I may have expected too much. So far, I have not been able to relate pure math to behavior, but I’m looking.

My methodology then is to aim for a general understanding of a field and only get involved with the details when they explain or clarify behavior to me. It does not seem to be very crowded on the path that I have taken.

Liam Uber: “It does not seem to be very crowded on the path that I have taken.”

Thanks for your sincere reply.

Coel is a professor of physics, and I of course did not want to put him down at his article. But, his idea that all math-axiom ‘must be’ empirical is totally wrong, although most (almost all) known math-axioms are able to make contact to the real world. I am quite surprised that no mathematician making comments at Scientia Salon. I will owe you an explanation for the above statement if I cannot give you a solid example, and here it is.

A new number system:

Axiom one: 1 + 1 = a; a can be any ‘nature number’ defined by the textbook number system.

When I choose a = 3, then 1 + 1 = 3

Axiom two: all theorems in the real-number (textbook) system is valid, applicable on this new number system.

Axiom three: a + 1 = b, b + 1 = (a + 1) + 1 = a + (1 + 1) ;

If a = 3, then b + 1 = a + (1 + 1) = 3 + 3 = 6

{(b + 1) + 1} can have a few different answers

{(b + 1) + 1} = [b + (1 + 1)] = [b + 3] = [6 + 3] = 9

{(b + 1) + 1} = [(6 + 1) +1] = [7 + 1] = 8

Axiom four: if a formula produces more than one answer, the largest one is the answer. [note: this axiom ensures that every function has only one value.]

I just made up the above number system, and obviously it will be a mess; but it is still a well-defined number system. After a bit work, we can find many wonderful and strange properties while they might be useless for our daily applications. But, by all means, this is a genuine ‘number system’. In fact, it can be proven that this new number system is ‘isomorphic’ to the nature number system. This is a big issue, and I will not go into any deeper here.

Hi Tienzen (Jeh-Tween) Gong,

Thanks, that is very helpful. It is becoming clear to me our minds work differently when engaged in different disciplines. The vast majority of us may suspect this but we don’t deal with it as a reality of life. The empirical evidence is mounting; each one of us is constructed differently and so significant differences in function appear inevitable. This leads to an inability to agree on any fundamental question, especially amongst experts.