“To do mathematics, one needs a pencil, paper, and a wastebasket. Do do philosophy, the pencil and paper will suffice.” – Author unknown

In school, we were all taught about the “scientific method”, wherein an idea (hypothesis) is tested by experiment. The hypothesis is invalidated by any single failure, whereas a successful experiment does not necessarily validate the hypothesis. Yes, this is important to science, but this is merely one part of the “toolbox”, just as an instrument is only one part of making music.

What is a scientist?

A scientist is a person that, first, is not disturbed by saying “I don’t know”, and second, who is willing to try to find out, even (especially) if nobody ever has: to the point of being actively engaged in the search. This could be classifying butterfly species in the Amazon, or smashing subatomic particles together and analyzing the results. I am a medical doctor: a technician. Admittedly, a highly-trained one, involving the operations of a very complex machine, but I depend on actual scientists to give me the information I need. (Note, some medical doctors are scientists as well.)

The “I don’t know” is part of what differentiates science from, for example, religion.

How does science operate?

The nuts and bolts of scientific research rely on the “scientific method”, referred to above. Formulate a hypothesis, test it against experiment (that is, see if the hypothesis fits reality), and reject the hypothesis if it does not. The more detailed the hypothesis, the more it fits with reality (ideally, better than the previous theories), and the more predictions it can make regarding future experiments, the more it is accepted. It also must be able to be replicated by others. The philosopher Karl Popper added this: for a hypothesis to be even considered scientific, it should be falsifiable; that is, it is incumbent upon the theorizer to formulate, at least in principle, an experiment that would prove the hypothesis wrong! This may sound counter-productive, but it is not. For example, I could hypothesize that there is a thriving civilization under the ice of Jupiter’s moon Europa. This does not run counter to any laws of nature that we know of, but according to Popper, it is not scientific, because there is (at present) no way to prove it wrong.

Carl Sagan famously said “Extraordinary claims require extraordinary evidence”. The more well-established a scientific idea is, meaning well-supported by evidence, that more difficult it will be to overturn it (supposing that it needs to be overturned!). The number of true paradigm-changing revolutions in science are few: Copernicus, Galileo, Newton, Dalton, Boltzmann (more on Boltzmann another time), Planck, Gödel, and of course Einstein. ^{1  2}

One point to clear up: the difference between a hypothesis and a theory. Common English uses them interchangeably (and so, alas, do scientists at times). A hypothesis is simply an idea that you are going to test. A theory is a hypothesis, together with supporting evidence, often with a mathematical foundation, that has been shown by experiment to conflict with no known laws of nature (which are themselves well-established theories!), and makes predictions that are at least as valid as can be made without the theory. Thus, the “Theory of General Relativity” is not Einstein’s mental noodling on the subject of spacetime and gravity: it is a whole boatload of very complicated math, long experimental verifications of expected physical outcomes, predictions of future experimental results, and ramifications into other fields of study. Which brings us to…

The Benatar Principle

All of science is provisional, to a greater or lesser degree. Some theories are so well accepted and verified that they are referred to as “laws”; well, Newton’s Law of gravitation held sway for almost three hundred years, until it was supplanted by General Relativity. General Relativity has been the tried-and-true means of understanding the relation between space, time, and gravity,

but its time to step aside may come as well. Science moves slowly at times, there is inertia, egos are involved ³, but science ultimately follows what I call the Benatar Principle: “hit me with your best shot”. ⁴ This is the self-correcting mechanism of science, which is the single most important principle in science, and the other vast difference between science and religion.

Is mathematics a science?

You will note that I put Gödel in the list of paradigm-changers. Gödel was a mathematician (a logician, to be precise), who, in the 1930’s, pretty-much-literally pulled the rug out from under modern mathematics with his Incompleteness Theorems. ⁵ The relationship between math and science is an odd one. Math is the indispensable tool of science, yet math, to be successful, does not need to conform to physical reality at all! In fact, most of math does not, but in the places that it conforms to reality, it describes that reality (in the words of Eugene Winger) “unreasonably effectively”. ⁶

Short answer: I don’t really know. Some believe that math is a process of discovery of pre-existing concepts, existing independently of discoverer, or even language (“mathematical realism”). Others believe we invent math as we go, that it has no independent reality of its own (“mathematical empiricism”). I will happily leave these questions to the philosophers!

 

 

1. And a few more
2. You will occasionally hear a modern scientist, or non-scientist, with an idea that seems outlandish, say “they laughed at Galileo too!” The difference is, they stopped laughing at Galileo.
3. It has been said that science progresses “one funeral at a time”.
4. In fact, nothing makes a scientist happier than to prove another scientist wrong! Sometimes they demonstrate all the maturity of middle schoolers on a playground.
5. More on these another time, as well
6. “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, article, 1960.