Have you ever been in an argument with someone online about the number of holes in a straw, and then you decided to “win” the argument by appealing to mathematics, saying that the field of topology proves you right? Well news flash buddy, you’re wrong.
Mathematics, when applied to the natural world, is not like the natural sciences. You can say that science proves we should take care of the planet, etc. but math just doesn’t work that way. The difference lies in the fact that mathematics typically models the material world - it is not built up from facts that are true in our world, but rather, facts that are universally true following some assumptions.
Consider the straw argument. While topology does say a straw has a single hole, it is intellectually lazy to say that settles the argument. That is because the mapping of the problem - the number of holes in a straw - to mathematics is largely what is being argued. The fact that by convention topology defines a hole a certain way does not mean that that is how the word “hole” is usually used, or that the convention in topology is the best one (you could just as well define a hole in topology as a pair of holes, for example). To say that the topological definition of hole is the only valid one is to say it is impossible to dig a hole in the ground. If you’re going to appeal to mathematics, you must show that the mathematical model you are using for the situation is relevant to the argument at hand.
For another example, the other day I was watching YouTube shorts when I came across a video involving a game show with the same format as Friend or Foe. As a quick recap, after earning money as a team, players are given a dilemma - they can secretly choose to be either friends or foes. If both players choose foe, neither gets any money. If one player chooses foe and the other friend, the foe-picker gets all the money. If both players pick friend they split the money. In the short, one player boldly announces that they are picking foe, so the other person should pick friend, and then they will split the money with them afterwards. In a twist ending, both players end up picking friend.
A huge number of comments said that what the non-“I’m picking foe” player did was game-theory optimal. Someone pointed out that technically speaking, their model isn’t accounting for the fact that human beings are spiteful, and place a value on effing a foe out of money. Commenters were quick to say game theory does not concern itself with this, as it is not rational.
What gobbledygook! What humans place value on is entirely arbitrary - the mapping of reward and punishment in game theory is entirely outside of game-theory itself, and is the realm of pyschology. You cannot simply say “wanting to earn money is rational and wanting to prevent an enemy from making money is not, because game theory”! And on top of it all, choosing friend in this situation is only technically optimal - if you know your opponent is going to pick “foe” nothing you do can possible give you money or not (and in the situation where you know your opponent is picking foe, humans, being spiteful, would place value on preventing them from walking away with all of the money). On top of all of this, picking foe for either player would have netted them double the money.
As an aside, I asked Gemini to explain weak Nash equilibria to me, and it entered an infinite loop of giving examples of game matrices and saying they do not have a weak Nash equilibrium. Ironically, the very first example it gave was the wikipedia example for a game with a weak Nash equilibrium.
Anyways, my point is, don’t just be going around doing a balk saying you are right because math like that. You have to actually show that your mapping of reality to math is compelling, otherwise
you might as well be saying “I’m right because if you multiply six by nine you get forty-two.”