So what I had gathered is that maths was for the most part discovered, meaning that it would exist regardless of what we do. But does all forms of math fit this? For example: are matrices something that were discovered or something that is governed by man made rules?
>>8093701
axioms are invented
the resulting objects and their properties are discovered.
Math wouldn't exist without us because our logic is not universal, it only makes sense to us. And that's what we use to do math.
>>8093701
Philosophy is just structuring and formalizing in natural languages.
mathematics are about formalizations of your speculations (which you form from your desire to see things that you experience [the empirical world, once you chose to objectify what you feel] through induction, as similar or dissimilar) to the point that you have a structure more formalized than your speculations structured in natural languages.
Logic is just a the formalization of your speculations about *validity of inferences*, so here logic is a formal part of mathematics.
It turns out that plenty of mathematical structures are cast into some formal deductive logic (like set theory formalizes your structures of numbers).
I meant your usual set theory cast in FOL. Set theory is just a structure too and it turns out that you can interpret a part of this structure as some kind of numbers.
Science is just claiming that your formalized structures (in formal languages or not) gives you access to some *reality*, more or less hidden with respect to what you are conscious of[=the empirical world, once you choose to ''externalize, objectify'' what you feel].
Same thing for the religions which go beyond empiricism [=claiming that you feel and think is **not** enough from which you choose to dwell in your mental proliferations].
Some mathematicians, typically Brouwer, think that mathematics should, equally to the speculations (however formalized) of the scientists, talk about the empirical world. So typically, your formal symbols are real entities: these entities belong to some world and they connect or not back to the empirical world.
to be clearer, the symbols are names of real entities and, since you begin always from the empirical world, this world constrains you on the creation and usage of these real entities. then these real entities can or cannot belong to some other world as well.
A lot of math, historically, came from "standard" (ie. euclidean) geometry.
So thats mostly discovery, because you can draw pictures and see how things work out.
(Our real world is approx. Euclidean 3d space)
However, once a certain formalism is established (like Cartesian coordinates, defining algebraic equations for plane curves, or axioms for deriving all known properties of geometry), one can twiddle around with that and invent new things.
Probably the most trivial example would be Euclidean 4d space, or more interesting, alternative axioms for geometry, in particular relaxing the parallel axiom in Euclidean geometry leads to hyperbolic geometry, which is a lot of fun.
pic related shows a bunch of lines going through some point and parallel to a given (thick) line, visualized in the poincare disk model of the hyperbolic plane.
>>8093716
>Math is just a language
>muh formalism
>muh structures
Gee, I wonder who could be behind this post. Couldn't possibly be someone who's had their long nose buried in a book for the past two thousand years and thinks reality is *literally* a linguistic *construction*, could it? No, no, it's just a coincidence. Nothing to see here.
>>8093701
The patterns that people see and attempt to describe with math are obviously discovered. You don't "invent" perception. Perception is not a "construction". So, while math is "invented", it is ONLY invented IN response to what is seen and perceived in the world. No math is invented in a vaccuum, regardless of whatever Shlomo >>8093716
and the rest of the Bourbaki and friends wants you to believe. Jews don't like to admit this, because their whole ponzi scheme depends on silly goy buying into their silly faith, but things *actually* existed and were perceived *gasp* before people could read and write. Wow! Imagine that!
>>8093716
Go outside.
>>8093716
>write a book, and decide to worship and study said book
>form a group
>selects for people who are "better" at "understanding" the book
>select for people who spend the most time reading, in-doors, arguing with other people who spend lots of time reading, in-doors
>group becomes progressively more near-sighted, progressively losing depth-perception
>group becomes progressively more skeptical of things that aren't words
>group refuses to understand things that can't be explained with words
>group becomes progressively more skeptical of "everything" and more confident about doubting reality that isn't what they are familiar with
>inbreeding increases frequency of said traits
>thousands of years later
>
>reality is a linguistic construction
>muh structures
>symbols may or may not be real
>I can't know if anything outside of myself is real
>muh everything is a language
Math is fundamentally understood, we just put a written system to it.
Ants do math to figure out how far away from the nest they are, it's pretty cool. They multiply their stride length by how many strides they take and know how far they've gone. This was tested by putting stilts on ants who had left the colony and noticing how when returning they would walk too far, past the colony. When you put the stilts on them before they leave the colony, the ants navigate away and back just fine because they're making this judgment with the stilts in mind.
Scientists use math which is expressed through the classical set theory that is ZFC.
Under scientific realism, the math done by the scientist is real, like electrons and photons are real (as some eigenvectors of their number operator).
This means that the axioms of ZFC are r.eal.
Now, one of the axioms is that there is a set (it is an axiom in the metalogic).
Now, where in the universe is this set? How do I observe it? measure it?
