In some of our exchanges during the past week, contributors have been discussing the nature of truth and the difference between “truth” in mathematics and “truth” in science. The assumptions in mathematics, are called “axioms” and are generally thought to be “a priori” constructs of the human brain. The assumption that a mathematical truth, such as; 2 + 2 = 4 is false, would yield a contradiction, so it must be true. The edifice that we call mathematics is a collection of deductions resting on a foundation of “self-evident” assumptions. The edifice is only as strong as the axioms, but since they are innate, they would seem to be independent of observations of the outside world.
In his Critique of Pure Reason, Emmanuel Kant distinguished between analytical truths–those, independent of observation and derived from purely logical deduction–and synthetic truths–those derived from experience and observation. He believed that the brain superimposes its a priori “axioms of understanding” on the external reality to make sense of it. He asserted that the mathematical axioms are analytical truths while the postulates of science are synthetic.
This “a priori” status of mathematics distinguishes its validity from the validity of the sciences. The truths in science are “a posteriori.” That is, the principles that describe nature are based on observations that are collected into a coherent body of knowledge. It’s conceivable that there is a galaxy in the universe in which Newton’s Law of Universal Gravitation does not apply, but it’s not conceivable that there is a realm where 2 plus 2 is equal to 5 or where 5 < 3. In the natural sciences, a mathematical model is derived from “assumed” principles. The model is used to make predictions that are tested against observations. When there is a discrepancy between a prediction and an observation, the theory is revised to accommodate the new observations. Scientists trust that this endless sequence of theories and revisions eventually converge to a final “truth,” as the gap between perception and reality shrinks.
An excellent presentation on the difference in the meaning of truth in mathematics and truth in physics is delivered by Richard Feynman in this classic YouTube video. It’s less than 10 minutes, so pour yourself a coffee or tea and enjoy it!
