What are theorems (lemmas, axioms, etc) exactly? I know what a theorem is, and I know how theorems are motivated; I also know what lemmas and axioms are. My question concerns something a little deeper. Are these theorems and mathematical tools we have merely a consequence of the way we've defined things? Are they discovered or created? Feel free to share some thoughts.
Well, the way we have defined things and the axioms we state using those definitions. We can have a mathematical structure in which the statement "through any point, not on a given line, there exist exactly one line parallel to the given line" is true and another in which it is false. Both! Theorems are created when we give basic definitions and axioms. We then dscover them when we prove them.
I would have to say they are created. The earliest notions of math started out as collections of discoveries and later rules to transform results. Then as the need for a system of math, axioms were developed to define what cant be proved and to then carefully construct theorems from the axioms. Euclid's geometry is perhaps the earliest and best example of a system of mathematics based on axioms and carefully constructed theorems. More recently, questions came up about the parallel axiom which resulted in the creation new geometries. I imagine that if a different parallel axiom were chosen then perhaps spherical geometry would have been created before Euclidean geometry.
Why are axioms self evident truth? That's as if assuming they're absolute from the get go. Humans seem to have systematically created all of this "mathematics" at the end of the day. For example, the way we define the addition operation (+). It means to literally take two precisely similar objects and combine them. "We" have defined it as such though, through... physical observation? Is math simply physics from the ground up then?
Axiom's don't have to be anywhere near the truth. They are just statements on which the structure is built and hence can't be proven using the structure. For example the euclidean geometry with its axioms holds only to an approximation in General relativity. (http://www.thefullwiki.org/Non-Euclidean_geometry) Mathematics at the end of the day is a tool (wow...I just spelled that troll...too much net...) to describe and solve real world problems (or was created to do so). An interesting read: http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html http://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences
I found this to be the most interesting portion of the darthmouth.edu article: After reading, I find mathematics has become fine tuned to the standard model when concerning physics, but also has its own independent motivation. Mathematics is more of a formalism, whilst physics tends to be concerned more with phenomena. Galileo's comment was striking, I'm not sure how he came to the conclusion or how he justified it though. Perhaps I should ask a philosopher.
They are not just a consequence of the way we've defined things. We've defined these theorems and tools so that they describe reality, and in that way these theorems are a consequence of how reality is. We create the theorems and tools - but the obedience of reality to them is a discovery.
When people talk about "describing reality" there are two ways this can be interpreted: describing the observable universe, and describing mathematical realities. Banach-Tarski for example describes the latter not the former.
If "we've" defined these theorems and tools so that they describe reality, then clearly "we" are describing reality with our own perceptions (which happen to mostly be physical). So how can we say these theorems are a consequence of reality when it is our perception of reality that makes them real. Can you say with confidence that mathematics is completely objective in this sense? Probably not. I agree with this.
You quote my previous post and then ask this? I didn't say that axioms are "self evident truths" because I don't believe any such thing! "Axioms" are part, along with definitions, of what determine the particular logic system we are working with. I know many different ways that addition is defined in different systems. You are basing your questions on things that just aren't true.
That any axioms or convention "describe reality" is still a choice on our part. We could take a new set of axioms that do not "describe reality" and derive new truths from them, that are just as true. Math is beyond "describing reality."
Yes, it is a choice on our part. But I don't think that there's much interests in axioms that do not describe reality in some way or another. Virtually all the math you learn in university is invented in order to understand something better, and that something is usually very closely related to the reality. Also, people get too hung up on the entire axiom process. Axioms are a way of presenting mathematics and to see whether everything is sound. It is not something to actually use in doing mathematics. Throughout history, you always see the same trend. Some mathematician does some research and finds some cool results. And only then does he think about which axioms he should use. People have worked with calculus for hundreds of years, and only very late in the game did they actually care about the specific axiom system. Same with groups, topology, differential geometry, etc. Axioms are usually chosen to abstract something that is present in reality and intuitively true (intuitive means intuitive for the experienced mathematician). While some mathematicians do invent some completely unreal axiom systems, they are rarely useful and are not studied a lot for that reason. Another source which shares my point of view on mathematics: http://pauli.uni-muenster.de/~munsteg/arnold.html
All I'm saying is that there are no mathematical truths independent of the axioms we choose to adopt, this is the direct response to the question I quoted. You can discuss our motivations behind adopting certain axioms or whatever, but that doesn't have much to do with the question the poster asked.
Well, I don't see anything suggesting that the poster is asking why we adopt certain axioms, or anything about limiting math to the physical world. The questions are, if theorems and tools in math are consequences of how we've chosen to define things, and if these theorems and tools are discovered or created. My opinion is that: 1.) yes, absolutely. 2.) foundations are created, we discover the results. I don't see how pointing out that we've picked axioms that are consistent with human perception is some kind of disagreement to what I've said, nor do I see relevance to the poster's question. He's asking about mathematical philosophy, not about using it to solve human problems.