Thanks! Luckily, I got a 5 in AP calc many years ago, so... I think I get it, sort of!
Unrelated: I had a new stress dream last night. My 6-year-old let all his school friends into our house when I wasn't home and they took all my watches and ran off into the woods with them. And my insurance refused to pay because it doesn't cover acts of children.
I woke in a state of panic. Someone please give me a bad psychoanalytic reading regarding this.
I'm firmly in the camp that math is invented by humans. It is a model of reality, not reality itself. Look up Gabriel's Horn paradox or the Banach-Tarski paradox.
Gabriel's Horn is a particular infinitely long straight trumpet and it is possible to prove that it has finite volume but infinite surface area. It requites an infinite amount of paint to paint the outer or inner surfaces of the horn (it has no thickness). However, because the volume is finite, it is possible to fill the horn with paint. Once you dump out the paint, have you not painted the inner surface?
The Banach-Tarski Paradox says that it is possible to take a solid sphere, break it up into parts, and then reassemble them into two solid spheres of the exact same size as the original. It is possible to prove this without much difficulty, but one cannot do this in reality because the disassembly would require an unaccountably infinite number of impossibly complex cuts.
Also, many aspects of mathematics are arbitrary, it could have been invented in other ways. Although the differences would mostly be inconsequential.
I tend to believe that math is discovered, not invented.
1 + 1 = 2 Is a basic truth. We certainly attached a language to it, but only to describe it.
Consider the sum of consecutive odd numbers. The result is always a prime number. How can that be invented?
You must have messed up something in your claim that the sum of consecutive odd numbers is always prime because it is not correct. The numbers 1 and 3 are consecutive odd numbers, but their sum 4 is not prime because 4 = 2 x 2. In fact the sum of any two odds is always an even number and therefore not prime, with the exception of 1+1 which equals 2 the only even prime.
Oops, I erred. Post corrected.
Happy international womens day everybody.
Gabriel's Horn is a particular infinitely long straight trumpet and it is possible to prove that it has finite volume but infinite surface area. It requites an infinite amount of paint to paint the outer or inner surfaces of the horn (it has no thickness). However, because the volume is finite, it is possible to fill the horn with paint. Once you dump out the paint, have you not painted the inner surface?
Conceptually, the Gabriel's Horn problem is an example with many practical applications in real-world physics. There are a variety of physical phenomena that converge in 3-dimensions but diverge with increasing system size in lower dimensions. The logarithmic divergence in 2D is particularly common, e.g. in the physics of liquid crystals.
And people wonder why I drink
I think you mean that the sum of the first n odd numbers is always a perfect square for any n. The sum of two consecutive odd numbers is not always a perfect square, for example 5+7=12 is not a perfect square. This picture demonstrates that the sum of the first n odds is a perfect square. Start at the lower left: 1 black square, plus the 3 white squares, plus the 5 black squares, and so on. It is pretty clear you get a perfect square.
By Anynobody - Vector version of Image😜roofwithoutwords.png, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=4268346
Yeah there are quite a few. @Dan S is an engineering prof. What discipline do you work in @CBM1590 and @Waltesefalcon, if you don't mind me asking?
