How do we know math isn’t wrong? Like it works but could it only work due to our perception?
In: Mathematics
You can think of math as a language and we use it to describe things in the physical world or to describe financial transactions or whatever. Over time many people agree on the basic definitions and properties of that language [ 2^4 = 16 and Sin(45) = 0.707]. So the “language” is correct if we all understand it and use it in a consistent way. How do you know that the meaning of a specific word isn’t wrong? Because most everyone who uses that language agrees on it meaning and the way the word should be used. Then when you get to the question of… does this equation correctly describe this physical phenomenon, you are in the realm of physics and we’ve all agreed on the language we are using to solve that problem in physics.
Math is an internally consistent set of rules. We use it for convenience and apply it to the real world, but math does not itself describe reality.
1=1 in math. 1 apple = 1 apple in math if we define an apple as a unit.
But in the real world apples are not equal. They can be different sizes, quality, types, etc.
So the math is correct as math. It is not correct when applied to the real world.
Math is a tool. It can be used incorrectly.
It can’t really be wrong. Math is a thing we created much like language. So you’re basically asking how do we know english isn’t wrong.
Axioms, axioms are statements which are unquestionably right, and the rest of maths are based on them
We don’t. At some point we need take things on faith. The good news is we don’t need much faith. For math most people use, there are a few ideas we take on faith called “Peano’s axioms” which are basic things like “1+1=2” and “I can add one to a number”.
We also have a LOT of experience to tell us that those things are true, but that only results in high confidence, not certainty.
For the not 5 year olds:
– check out Godel’s incompleteness theorem
– prove to me the concept of “previous observations are relevant to future events” without recursion.
The first thing to understand is that math is not a part of science. Math doesn’t rely on the real world to be right or wrong. It’s right or wrong on its own. Math is right if it is internally consistent, regardless of whether it is useful for something in the real world. What’s neat is that much of math *is* useful in the real world.
Basically, the way math works is you make up a few simple rules. You then take those rules and see where they lead the more you play with them. If you find a set of rules that does interesting things, you’ve found some interesting math. If you find a set of rules that can describe things in the real world, then you’ve also found something that may be useful, besides also being interesting.
Math is a language that is used to organize a series of definitions and logical statements. For example. If I had three cows, I could directly represent that I had these cows by drawing three lines: III. This is how people in many ancient civilizations would have done it. Eventually someone smart came along and said “Representing things with lines is easy, but it doesn’t work well with large numbers. Instead of using lines, lets use the symbol ‘3’ for your three cows. Then, if you have so many cows you need to count on your fingers 3 times, write it as ’30’.” The symbol ’30’ doesn’t have any direct connection to a bunch of cows, but the discovery that numbers are more flexible when written in this way (Hindu-Arabic numerals) helped people better express themselves.
The idea that 3 is equivalent to III is just a convention we all agree on because it helps us to communicate or organize our own thoughts better. It holds no fundamental truth, and the same is true of all math. We didn’t discover arithmetic/calculus/statistics/etc. We invented them. They are tools, and tools can be judged directly by how well they work. Many people who accomplished remarkable feats in the real world did so with math, so we know the mathematicians who invented that math did a good job.
Math is a language of representation. That being said simple mathematics can be proven with physical equivalents. Like for addition if you have 2 rocks and your friend has 2, together you have 4. If you draw 3 circles, and put 4 rocks in each, that’s 12 rocks. Have 12 rocks give each 3 people 4 rocks. For more complex maths, its just a combination of proving something simple and going from that. Other people say proofs and thats about it.
I’d like to point out Gödel’s incompleteness theorems.
They’re not exactly ELI5 friendly, but the gist of it is basically : “you cannot prove the correctness of a system from within it”.
This also applies to logic (math basically).
So we cannot use logic to prove logic.
Therefore, to answer your question : we cannot.
Edit: at least, not using logic. This leaves way to all sorts of emotional explanations and philosophical exploration, which many like to explore.
BUT
As long as the system (math) does not contradict itself, we admit that it is “correct enough” for our uses, and we choose to use it.
Plenty of mathematicians think that “math is wrong” if you mean that math is not physically “real” and only a construct of perception. In philosophy of mathematics, this school of thought is called mathematical anti-realism, and there are many different forms of it. Here’s a quote from the book *Measurement* by Paul Lockhart that might help you see what I mean about math being “wrong”:
*”The thing is, physical reality is a disaster. It’s way too complicated, and nothing is at all what it appears to be. Objects expand and contract with temperature, atoms fly on and off. In particular, nothing can truly be measured. A blade of grass has no actual length. Any measurement made in this universe is necessarily a rough approximation. It’s not bad; it’s just the nature of the place. The smallest speck is not a point, and the thinnest wire is not a line.*
*Mathematical reality, on the other hand, is imaginary. It can be as simple and pretty as I want it to be. I get to have all those perfect things I can’t have in real life. I will never hold a circle in my hand, but I can hold one in my mind. And I can measure it. Mathematical reality is a beautiful wonderland of my own creation, and I can explore it and think about it and talk about it with my friends.*
*Now, there are lots of reasons people get interested in physical reality. Astronomers, biologists, chemists, and all the rest are trying to figure out how it works, to describe it. I want to describe mathematical reality. To make patterns. To figure out how they work. That’s what mathematicians like me try to do.”*
Math is largely in our heads, but we can still know if the constructs that we use in math are true based on what we say the constructs are and how we say these constructs relate to others. As a very superficial example: if I define zero as “the number that keeps some number the same when you add it to some number”, then I know for a fact that n+0 = n where n is any number. This doesn’t necessarily have anything to do the objective rightness or wrongness of what “zero” actually means.
You might also like the book The Number Sense by Stanislas Dehaene as a good introduction to a field called “numerical cognition” if you are interested in the relationship between math and the brain/mind.
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