How do we know math isn’t wrong? Like it works but could it only work due to our perception?
In: Mathematics
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.
Math isn’t right or wrong, it just is. It’s a logical deduction from some basic assumptions, called axioms. For example, the theory of natural numbers develops from Peano’s axioms. Some parts of math have nothing to do with the real world and work with completely abstract things. We know that the parts that have something to do with the real world work, because they fit our observations.
Math also explains things that we cannot perceive which in turn has effects that we do perceive. i.e. quantum mechanics. That being said, our math isn’t perfect, we can’t fully understand everything via manipulating symbols, but we can understand most things up to a certain point.
Proofs. They start teaching you proofs in Geometry, well before they let you get anywhere near Calculus. Everything else is a “theorem”, a “conjecture”, or an educated guess.
Find a “good” teacher and ask why it’s good to learn proofs before you get to calc.
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.
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.
Simplified answer:
When you have a theory in maths or physics, you can usually prove that it is right by assuming that it it is wrong, thus that the opposite is true, then you start working on some equation using that assumption, until you inevitably stumble upon a oaradox. The paradox proves that the opposite of the initial theory is false, thus the theory is true.
Google something like proving that the root of 2 is irrational to see it done.
But if you mean the whole system…
Well… It just works. It replicates the real world. Imagine if you said that addition should be done before multiplication. You would end up with wring results compared to stuff in the real world.
I guess that is just a very simple answer, but someone might privide something more complex.
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.
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