I was just having this very old, neverending debate. I would like to have your opinion about this. It all started with geometry, but I think the argument extends to mathematics altogether. According to my friend, mathematics first come from experiment and thus belong to the category of physical models of the world. In my opinion, mathematics belongs to the ideal platonic world. I will first re-phrase my opinion since it is eaiser for me Considering that one can study mathematics without knowing anything about the outside world : I know of a blind geometry professor the physical world we use as a source of inspiration to choose the geometry we want to study, but we can invent (but see later) as many as we want, as wild as we can think of I think mathematics exist independent of the physical world. At the very least, mathematics are universal, a circle is a circle no matter if I am a french man, or an alien in another galaxy... He, on the contrary, argues that everything comes from experiment, and that just like other physical models of reality, geometry somehow belongs to physics. The mathematics we choose to study come from experiment, and are thus a reflection of the physical world The all debate actually seems to stem from the The Unreasonable Effectiveness of Mathematics in the Natural Sciences. I also want to push the two previous opinions to their extreme : One could consider that mathematics are just a toolbox. If a piece of mathematics serves no practical purpose it is useless, should be disregarded and not taught. I am even among those who think that mathematics is just like art, because I often experience deep esthetics feelings while reading a mathematical proof, intense as one can feel while contemplating a painting of listening to a beautiful musical composition. I do not want to develop to much at this stage and/or go into philosophical consideration, but earlier I would rather have said discover instead of invent since I think mathematics exist per se, before we know them, in the platonic world.