Math and How it Exists

What is mathematics, really? For me, it’s all just a bunch of numbers systematically arranged to be able to convey one thought or give some solutions to a problem. Actually, it’s more of using numbers and symbols in solving problems in life, or sometimes causing the problems by getting additional assignments or problem sets. After reading the book, it changed my view entirely.

                It started with a conversation of the author with a child. He asked her what a gazillion plus a gazillion would be, and she answered confidently: 2 gazillion. I had the same thought with the child, but then I realized I didn’t know what a gazillion meant. Apparently it stood for a very large number, so why answer 2 gazillion? Is there such thing as a 2 gazillion, I asked myself. This is the part of the book where I realized that I was going to see a whole new side of mathematics.

                The contents were mainly about how the philosophy of math changed over time. Honestly, I was not expecting that. But as I read, I began to question myself: Do numbers and symbols really exist? Reading the book made me thing real hard of that. Before, when I was still in high school, I remember imagining myself in a white space full of nothingness except for a huge number line in the middle, and I was situated at the 0 point. I looked front, and I imagined the numbers reaching infinity. I looked back, and thought that if I run towards the end, will I ever reach negative infinity?

                The author somehow saw through this, and he even was able to solve the parts of a 4D cube using a pattern he got from analyzing 1D, 2D and 3D. He solved 4D, but it doesn't exist in the physical world. It is mainly in human thoughts, and that is why we believe that it exists, like we think how numbers exist.

                He was able to make me wonder about math. This is what I like about the book, because math never made me wonder at all. Solving was tiring for me, but studying existence is far more interesting. Does math exist before humans? Can it exist without us? He even told an example, like if 1 dinosaur drank in a lake and then another dinosaur comes and drinks too, that would make 2 dinosaurs. That is math in the time of the dinosaurs. Imagine that! Match can exist without us, but we have only refined in the later ages, when dinosaurs no longer roam the earth. Math could have been here before humans, but it would not have any meaning to other beings like animals and plants.

                The author made me look at math from a Platonic point of view. For Plato, math exists independent of humans, like there’s a whole other universe where they can exist without us. Another view is through formalism where math is mainly statements that follow certain rules. Personally, I like the Platonic view; because I have this strong belief that math could exist without us, thanks to that dinosaur example.

                There happens to be some math myths that the author mentioned, and I’m glad he did so that my beliefs could be corrected. Turns out that the math we know isn’t the only math there is, and it is not true and the same for everyone. There could be another universe using math differently, and even in history, there are certain places that developed math independently, leading to it not being the same for everyone. I chemistry, 2 atoms of one element plus another 7 atoms of another element could result in 4 atoms of the compound. But because it seems that we have the same math as with everyone else people begin to believe that there is only one kind of math.

                The writer also discussed some interesting math puzzles, which was my favorite part because I really love puzzling facts. Is math discovered or created? Honestly, I have no idea. But for me, since I’m kind of in favor with the Platonic view, I think it could be that we discovered math by trying to acquire our needs. We needed to keep track of time, so we discovered a way to do it. It could also be that we created math. This is why I love philosophical puzzles; they are like Schrödinger’s cat, could be right and wrong at the same time or it is possible the answer is none of the above. Another puzzle is the concept of infinity. I love this, because infinity is there, at the same time it is not. It is in our thoughts, but we only have a concept of infinity, not infinity itself. No man could stick infinity into his finite brain. The meaning of an object and a process was also discussed. He was able to come up with a conclusion that an object is a slow process; a process is a speedy object. It actually made sense though, like when a stone is falling it is a process, but when time stops, it is now merely an object. There is also changed, and I like how the author discussed the change in the number 2 throughout time. The last puzzle is the existence of nonexistence. Sometimes we say that the limit does not exist, but what does it mean? Diving 1 over 0 has no definite answer… but it doesn’t mean that there is no answer, for there are actually an infinite number of answers.

                The growth of the philosophy of mathematics was described as a train on rails: it projects little by little towards the future. Part 2 of the book discussed the growth of math throughout the years.

                Honestly speaking, this was my least favorite part of the book. I know that it is important to know about the history, but there are some parts I find very hard to understand. Sometimes the philosophy is way over my level. I think I remember that Pythagoreans treat math as a way of life and are very strong in their beliefs. Plato started the Platonic view of math, and there were neoplatonists that renewed some of the old ideas. St. Thomas Aquinas was also a part of this, and what I can remember from him is that when you get rid of all mathematics from a mathematical object, you are left with the real object.

                Several names were mentioned, all of them contributed in building mathematics. My favorite part was when they discovered more about logic. At first, I could only think of logic as… logical. But now, I have seen a new definition of logic: it is the rules of computing machinery. I loved how Gottlob Frege pretty much attacked a lot of philosophers to prove that numbers aren’t just ideas. Also, he was a strong believer of logic. But then the Russell paradox moved him, leaving him speechless. He wanted to make a solid foundation for math, which he failed to do.

                There were other concepts of viewing math, like humanism and fictionalism. Fictionalism discusses the levels of existence in math, while humanism believes that we, the human race, are the beings that created the concept, which is totally opposite of Platonism. The author then discussed the humanists (modern and contemporary).

                After tracing the history, he then went back to his criteria and judged the philosophies. There were so many kinds, and not all fitted his criteria perfectly.

                The author did an amazing job in collecting all those facts about the people who tried to contribute to the philosophies of math. I particularly loved the way he wrote naturally and added his opinions in between, and I especially love it when he boggles my mind. The only problem for me was that there were times that I could not understand what he wrote, maybe it was way beyond my comprehension. Unlike most math textbooks, this one caught my attention.

All in all, math has such a tiring history. I thought it was all Egyptians and Greeks fumbling around with numbers, but then these men with deep ideas came along and tried to create the perfect meaning for math. Of course, this is hard to achieve. For me, math existed way before us, and right now, most of the math we study is usually logical. Of course, here comes infinity which is somewhat not logical, meaning math cannot have just one philosophy governing it. He concluded that math is not mental or physical; it is social. We can’t feel it but it is there.