When people want to defend a transcendental world view (particularly against science), they often bring up mathematics.
“How can you scientifically prove that two plus two equals four?” they ask.
Since you cannot, mathematics must be transcendentally true. However, this shows a misunderstanding of how one can use science to understand aspects of philosophy. Using the scientific method is not the only to prove something is factual (“true” in Nietzsche’s “uninteresting” sense in his “Truth and Lies in a Nonmoral Sense”). If we can show it is natural for the brain to process the world in such a way that we can make statements such as 2+2=4, then we have a science-based (though not “proven” scientifically per se) explanation for math, meaning we do not need a transcendental explanation for it. A science-based epistemology will do just fine.
Math is the abstract expression of relationships in nature. Words are sounds representing conceptual categories, which are derived by observing many objects and placing those objects with certain similarities into categories. Take the Bactrian camel. We categorize camels as either Bactrians or Dromedaries, because the Bactrians have two humps, and the dromedaries have one. All the Bactrian camels more closely resemble each other than any one resembles a dromedary, or any other animal, for that
matter. So we categorize two-humped camels as Bactrian. But are Bactrians in fact identical? No, each one is different—we erase the differences so we do not have to create a different category for each individual object in the world, which would be very cumbersome (this, despite the fact that we do oftentimes give an individual its own category, as when we name our pet dogs and cats, because we become so familiar with them that they become more individuated in our eyes). Our brains, to be more efficient, conceptualize. If brains did not do that, the owner of the brain would not recognize that the cat that ate a member of the group was similar enough to the approaching cat that it would be prudent to try to escape. This is why and how vervet monkeys can have a different call meaning 1) big cat, 2) eagle, and 3) snake, which each result in different responses.
The same is true of the one who needs to eat: one must be able to recognize what is not-food and what is food. To eat, one cannot have to relearn this information each time. Life was much simpler for one-celled predators: eat anything that moves (or, more accurately, whatever binds to the outside of the cell, meaning there is a kind of conceptualization even at the chemical level in the surface proteins). Those who could not make these judgements about the world and create concepts such as these would have died either from eating something poisonous or from being eaten. Any animal that could not make proper judgments regarding the reality of the world they were in would not have been able to survive. Those who were better able to make those judgements would be able to survive even better. I give as evidence a human population of over 7.5 billion people at present.
Due to the complexity of our brains and our use of language, humans are able to create more conceptual categories than any other animal. Further, those categories can overlap, and they can exist in nested hierarchies. The Bactrian camel is simultaneously a camel, in the camel family (which includes the humpless llama and its relatives in South America), an herbivore, a mammal, a vertebrate, an animal, and alive. Thus a Bactrian camel shares similarities with other herbivores—elephants, rabbits, buffalo, manatees, etc.—in that they all only eat vegetation. We call it a mammal because it has hair, is warm-blooded, and feeds its young milk, just like platypuses, whales, koalas, and leopards. It is a vertebrate because it has a backbone and an internalized skeleton, like fish, birds, reptiles, and amphibians.
At this point, you’re probably wondering what all of this has to do with math.
“Two” is a conceptual category. It is necessary to keep track of group members (we are a social species after all), and to make proper divisions in a group (as chimpanzees share meat from a kill). It would also be useful information to share if one is hunting or gathering. “Yes, there are two of them. We need more hunters.” “Two” is a conceptual category in the same way as “Bactrian camel” is. If we have two humps on two different camels on two mountains with two rivers flowing from the two mountains, then the relationship among humps, camels, mountains, and rivers is “twoness”—a conceptual category we designate by using the word “two”. Once we eliminate all the differences among these examples, the only commonality they have are number. The sound “two” may be an arbitrarily chosen sound to represent this concept, but the concept, and the fact that a word exists to represent it, are not arbitrary. Number is a high-order abstract concept.
The rest of the equation is relationships. Relationships are inherent in nature. Math describes nature so well because all of nature is relationships, each object has and is in a relationship with other objects (objects here including pure, substanceless energy). “Equals” is such a relationship. If there are a number of objects that we call (in English) “two”, and we have another number of similar objects that we would also call “two” due to the number of objects being equivalent, then we have a number of objects that we designate in our language as “four”. If “four” represents this many objects: * * * * , and two represents this many objects: * * , and 2=2, then 2+2=4. A transcendental explanation is not needed in the least. All we need is a proper understanding of how concepts are formed in the brain, and we can learn that through cognitive psychology, which is science. Thus, while science cannot prove using the scientific method that 2+2=4, while 2+2 cannot equal 5, and never can as long as we use the language and notation as we presently do, I have shown that a proper understanding of science can help us use philosophy to understand the source of mathematical statements as non-transcendental.
Further reading: Diaphysics