Mathematics is not a Science

Have you ever wondered what makes a study scientific? Well so did Karl Popper, undoubtedly one of the greatest philosophers of the twentieth century. If you’ve done even an ounce of philosophy, then you would know how difficult it is to not disagree with someone else. Popper’s ideas, however, seem so basic, so indisputable, that they have become one of those few rare gems in philosophy that almost no one disagrees with.

Many of the cornerstones of modern science were produced in the nineteenth century: relativity, psychoanalysis and quantum mechanics, just to name a few. Karl Popper observed the intellectual titans around him at work and wondered how their methods differed. In the process of uncovering this, he ended up teaching us a great deal about the nature of knowledge and why (he didn’t say it like this but let’s run with it anyway) mathematics is not a science.

Karl Popper was fortunate enough to be born in a time when both Einstein and Freud were breaking new grounds in their respective fields. He noticed that these great minds used very, very different methods. For example, Popper noticed that Freud could magically make anyone’s life story work into his theory. Evidence to support Freud’s theory seemed to be absolutely everywhere! Einstein, however, was looking into the far future and predicting what would happen. To Popper, Einstein was a real gambler, if his predictions did not line up with what happened in the future, his theory would go kaput. Since Freud could always interpret the past differently, he could make any data work into his theory and predict events in the present. Popper realised the difference between what Freud was doing and what Einstein was doing, and labelled Einstein’s work as science and Freud’s as pseudoscience.

To Popper, pseudoscientific methods only served to confirm beliefs whereas scientific methods are always out to get you, they aim to disprove a theory; disproving something, looking for evidence that goes against what the theory predicts is how one gets closer towards the truth. We would never have found black swans if we just kept looking for the white ones. In a single sentence, this is what Popper realised: pseudoscience confirms, whereas science disconfirms.

Scientific methods, however, as brilliant as they are can only be applied to things which are falsifiable. Nothing which is unfalsifiable can be tested by science, by its very definition. I would not know if an undetectable monkey is dancing behind me right now, because by its very nature it would be undetectable. Science does not bother with these sorts of questions. This is both a good and a bad thing. Many of the beliefs we hold so near and dear to our hearts (a practice we really shouldn’t do) cannot be backed up by science. Thankfully we have other branches of knowledge which (after much help from science) do at least attempt to answer these questions. But this requirement of being falsifiable for being considered for scientific analysis has some very interesting consequences. Since we can only ever disconfirm a theory and never confirm it, we never truly know anything for sure. All the data that we have gathered might sit well with a particular theory, but tomorrow we might observe something which sends the existing models and theories for a toss. We cannot ever be sure if we what we know is true or not because we get to the truth by ruling theories out. If we can’t rule a theory out, it does not mean that that theory is correct. If I can’t find evidence for black swans, that does not mean they might not exist.

But that is only true for science. Mathematics is much, much more beautiful than that. In mathematics you have proofs! A definitive argument which proves that a statement is true or false. This is totally unheard of in science. A proof in mathematics is one mathematician holding another mathematician’s hand and taking him from one indisputable truth to another and showing him that one truth implies another, and another, and so on until you reach the desired result. These foundational truths are called axioms, or they might be other truths which can be derived from certain accepted axioms.

Because of proofs in mathematics, and the existence of definitive proofs, mathematics is by far the furthest thing from science than any other field of study (yes, even more than pseudoscience!).

Just because something is not a science does not mean it’s not helpful, though. Just look at psychology; it was used as the prime example of pseudoscience by Karl Popper, but it has grown into much, much more than that. Psychology today takes us on epic adventures through our minds and helps us understand the motivation behind the actions of individuals. Another example would be yoga. Science is just beginning to understand the benefits of meditation and yoga, and is not even close to explaining why these practices have the benefits that they do. I am obviously not telling you to believe in everything you see and hear about. Rather, you should approach seemingly non-scientific things scientifically. Test and question everything. If it works for you, well then keep at it! And if it doesn’t, abandon it.