#math philosophy

“[Y]ou need only knock upon the wall of the storeroom and say, ‘One, one thousand, moon, one thousand, gold, ten thousand, nine.’ Instantly the seeds will become counted, …”

- from Chapter Five: Svetlana and the Seal’s Skin

Sorry I just um forgot about this.

Some aspects of math are discovered and some are invented. There are inherent structures of, well, things. Logic is largely sound (though I'm not willing to say inherently sound since like. Paradoxes definitely exist *cough axiom of choice*). These are the sorts of things which are discovered to be true.

However all these inherent truths are based on other things we just kinda decide are true. These are our axioms and definitions. That's what axioms and definitions are, things we decide make sense and result in nice math so we say okay this is a thing we call a ring or the real numbers are a set which is the completion of the rationals (I would argue natural numbers straddle the line between inherent structure and semi-arbritrary object). So I would consider our decisions to accept axioms and definitions is inventing math, from which we discover further properties.

I was never a fan of mathematics. Maybe it was due to the way it was thought to me in middle and high school. I always saw mathematics as an obligation, rather than understanding what truly lies behind it. The fact is that math is all around us; whether we like it or not. The Guardian did a piece on a French mathematician by the name of Cedric Villani. The way Cedric talks about mathematics was really passionate and basic at the same time. Explaining all the misconceptions of math and how understanding it makes us see the world through a different lens. 

Mathematics is considered as a social activity. It is also regarded a branch of science in a way, having a reaction to biology. Since biology is the study of life, and since mathematics is a feature of life, then we can say with confidence that mathematics is philosophy of science. The philosophical side of mathematics is due to the fact that math is about concepts and logical reasoning. Mathematical discoveries are discoveries we make on life itself. Since mathematics takes extensive reason and logic to comprehend and apply it, we can therefore clearly see its’ relation to philosophy. Plato stated that mathematics were a way understanding more about reality. He was a fan of geometry, which we believed was a way of unlocking the secrets of the universe. If we were to consider mathematics as a branch of philosophy, it would then fall under Epistemology - study of how we know things. 

Since truth is considered to be a matter of correspondence between our thoughts and language, and since words are pretty much ideas formulated in a way for our society to set a proper order; and since mathematics are true, its’ features must be a part of our daily lives. The idea of Mathematical Idealism is defined as the idea of thinking that there are numbers in the same way there are chairs. Which is to say that mathematics are a part of our life and not a mere creation of our minds. 

Gottlob Frege & Kurt Godel focused on the relation of mathematics to language and the formation of language, as well as its’ roots. When language interacts with the world, it does so by referring to objects; we can then conclude that numerals too are objects. Another approach to math’s relation to our daily lives was through the understanding of Structuralism : learning mathematics (or any other topic) through patterns and by pattern recognition. Structuralism also states that mathematical knowledge comes from experiencing various concrete or physical patterns; we can then see numbers as not individual objects but as a part of a certain structure. 

I find it beautiful that the "normal" to a line is the negative reciprocal, and the force "normal" in physics is the force perpendicular to the to the surface of contact.

The "normals" are the opposites. They're what push up against gravity's pull down. 

Normal in math and physics is not what you see when you look at the problem. It isn't smacking you in the face yelling "LOOK AT ME!". Normal in math and physics is just there to oppose the initial aspects.