HS #129 2026.4.9
Kinship of Mathematics and Science
In May 2024, we discussed whether mathematics is invented or discovered. (Remember?) Now, we explore another long-battled question about mathematical knowledge: Does one learn mathematical truths from observing the natural world, or are we born with mathematical knowledge inside of us?
Kant believed that unlike scientific knowledge of the natural world gained via experimentation, mathematical knowledge “rests upon no empirical grounds.” He claimed we know mathematics a priori (prior to experience); we could discover it in a dark closet without ever seeing the world.
I respectfully disagree.
It seems to me that mathematics is initially learned via experience. It took Bertrand Russell and Alfred Whitehead ten years and 379 pages of logic to prove that 1+1 = 2, but the rest of us know it by counting our fingers. Given a circle, the length of the circumference divided by the length of the diameter always gives the same ratio, which we call pi. How do we know? Experience. Indeed, I send my math classes to athletic fields with ropes of different lengths where they pace off the circumference and the diameter. Always get pi. Another example: Why do we know that the order of adding numbers doesn’t matter (7+9 = 9+7)? Because we’ve tried it many times and it works. It's an axiom of the real numbers; we are convinced of its truth via experience. Why do I know that five is a prime number? Because no matter how I try to divide this many dollar signs $ $ $ $ $ into equal piles of size greater than one, I can’t.
Why do we know that two points on a plane determine exactly one straight line? Having seen straight lines, we can visualize it. Conversely, why can’t we picture a four-dimensional object? (Salvador Dali came the closest – see his Crucifixion (Corpus Hypercubus).) Four-dimensional objects are just as math-legit as three-dimensional objects, but we don’t have the benefit of seeing them, so we find them difficult/impossible to visualize. Similarly, it’s impossible to imagine the next color in the rainbow – the ones we’d enjoy if our eyes could see ultraviolet or infrared light. We humans have an embarrassing lack of imagination!
So, as with experimental science, it seems we learn foundational mathematical truths from experience. However, once those initial truths are determined, then mathematics proceeds quite differently from science.
A coin illustrates the difference. Having observed that both sides of a coin are HEADS, we DEDUCE that when flipped, it will always show HEADS. Similarly, having all the information they need (their axioms), mathematicians proceed deductively, using logic to jump from one conclusion to another. They are confident that “proved true” means it IS true.
On the other hand, if without looking at a coin, you flipped it 100 times – always getting HEADS, then you might – via induction – become convinced that it is a two-headed coin, even though you will never know for sure. Similarly, scientists never fully assume the truth of their foundational assumptions. Instead, they acquire knowledge inductively by observing regularities and patterns in nature and then draw hopefully-correct conclusions – a process called the scientific method. Karl Popper realized that, like the tossed coin, scientific experiments can never prove anything true. One becomes increasingly convinced, but never certain.
Indeed, consider these historical conclusions of physics: i) energy is conserved, ii) mass is conserved, iii) an object’s length and mass doesn’t change, iv) velocities in the same direction can be added. The truth of all of these were challenged by the brilliant mind of Einstein, and now have now been shown to be wrong via experiments.
Mathematics, historically, also has had incorrect conclusions: i) every line has another which is parallel, ii) every number can be written as the ratio of two integers, iii) if something is true, it is possible to prove it true, iv) if you take some away, you have less. How simple and obvious – but all shown to be wrong! No experiments were needed, just pencil and paper through mathematical argument.
Surprisingly, mathematics and science are also similar in challenging a fundamental truth. Even though they proceed by disparate methods, physics (via Schrodinger’s Cat) and mathematics (via sizes of the infinite) agree that, paradoxically, there exist things which are neither true nor false. That is, the world can’t be neatly divided into true things and false things. Reeks of postmodernism!
How neat! Mathematics and science, each in its own way, reveal that our world is rich with wonder and surprise.
