Well, I'd like to challenge your assumption that maths at its core is inconsistent. I'd say rather, that our attempts at making an axiom system for maths leads to paradoxes rather easily, in a surprising way.
And that only happened when we started to feel a need for a single axiom system so general that you could create a foundation for the whole of mathematics with maybe a dozen or less axioms.
Sort of like - it's really a problem with providing a "Theory of Everything" for maths.
Mathematicians did geometry and group theory and worked with numbers for many centuries and millennia with no issues at all.
Euclid's axioms still are valid today, with only minor tweaks, axioms that Euclid left out.
How much of Greek science is valid today?
Also - though we can't prove that Peano axioms are consistent (a simple system of axioms for numbers) - I don't think anyone expects to derive a contradiction from those axioms.
So - what caused problems was Hilbert's program, to create a single unified foundations for the whole of mathematics. And that was motivated in turn by Cantor's paradoxes and the complexities of the set theory - which in turn came out of Fourier's work originally - to do with practical work on heat flows in solids.
So - is surprising that something so practical would lead all the way to such complex intricate maths and eventually to the paradoxes.
Does that mean that, e.g, there is something paradoxical about heat flow as such?
Or something paradoxical about the maths itself?
Or is it perhaps more to do with how humans try to understand these things?
Some mathematicians I've discussed this with think of maths as just like a physical theory - that we can expect it to need tweaking and it's no surprise that our first attempts hit inconsistencies until we worked out how to do it properly.
Is that right? Seems a bit strange to me, why would maths be like that, you can understand physics being like that, but maths?
I just see questions here, and not much by way of answers.
However - a bit of background - it is easy to create an inconsistent set of axioms. E.g. just add 2 + 2 = 5 to Peano's Axioms, that's an inconsistent system.
So - it's not such a big deal as that to have an inconsistent system of axioms, and doesn't prove that "all maths is inconsistent".
What happened is, rather - that there are some axioms that are inconsistent in surprising ways, that weren't expected - especially Frege's system.
And those axioms were originally, thought, by their creators, to be good candidates for an axiom system for the whole of mathematics. But - obviously are not since they are inconsistent.
Then Godel showed that Hilbert's aim to create a provably consistent set of axioms for the whole of mathematics could never be achieved, or even, one powerful enough to include arithmetic and Peano's axioms.
That's far short from saying that maths as a whole is inconsistent.
You can start with more restricted systems of axioms, not general enough to capture the whole of mathematics, but powerful enough to contain just about anything that human mathematicians are interested in.
You can never show conclusively that it is consistent, but you believe it is.
So that's what mathematicians do, that's the basic idea of the mathematical systems such as ZF, ZFC, etc.
By Godel's incompleteness theorem, any powerful enough system of axioms can never capture all of maths.
If you've described it clearly enough so that it's totally clear how theorems are proved - then by his ingenious methods, that leaves it open to processes of adding new axioms to the system, which a mathematician can see must be true, but which are not included in your original list of axioms (it also leaves you open to the possibility of adding in the negations of those new axioms).
But you can then expand your axiom systems as needed in creative ways if you need to go beyond their limitations.
So - it means maths has to be creative, and never ending. And we can never know for sure that it is consistent once it reaches a certain level of complexity and power.
But doesn't have to be inconsistent.
Perhaps same is true of the universe? I don't know.
And that only happened when we started to feel a need for a single axiom system so general that you could create a foundation for the whole of mathematics with maybe a dozen or less axioms.
Sort of like - it's really a problem with providing a "Theory of Everything" for maths.
Mathematicians did geometry and group theory and worked with numbers for many centuries and millennia with no issues at all.
Euclid's axioms still are valid today, with only minor tweaks, axioms that Euclid left out.
How much of Greek science is valid today?
Also - though we can't prove that Peano axioms are consistent (a simple system of axioms for numbers) - I don't think anyone expects to derive a contradiction from those axioms.
So - what caused problems was Hilbert's program, to create a single unified foundations for the whole of mathematics. And that was motivated in turn by Cantor's paradoxes and the complexities of the set theory - which in turn came out of Fourier's work originally - to do with practical work on heat flows in solids.
So - is surprising that something so practical would lead all the way to such complex intricate maths and eventually to the paradoxes.
Does that mean that, e.g, there is something paradoxical about heat flow as such?
Or something paradoxical about the maths itself?
Or is it perhaps more to do with how humans try to understand these things?
Some mathematicians I've discussed this with think of maths as just like a physical theory - that we can expect it to need tweaking and it's no surprise that our first attempts hit inconsistencies until we worked out how to do it properly.
Is that right? Seems a bit strange to me, why would maths be like that, you can understand physics being like that, but maths?
I just see questions here, and not much by way of answers.
However - a bit of background - it is easy to create an inconsistent set of axioms. E.g. just add 2 + 2 = 5 to Peano's Axioms, that's an inconsistent system.
So - it's not such a big deal as that to have an inconsistent system of axioms, and doesn't prove that "all maths is inconsistent".
What happened is, rather - that there are some axioms that are inconsistent in surprising ways, that weren't expected - especially Frege's system.
And those axioms were originally, thought, by their creators, to be good candidates for an axiom system for the whole of mathematics. But - obviously are not since they are inconsistent.
Then Godel showed that Hilbert's aim to create a provably consistent set of axioms for the whole of mathematics could never be achieved, or even, one powerful enough to include arithmetic and Peano's axioms.
That's far short from saying that maths as a whole is inconsistent.
You can start with more restricted systems of axioms, not general enough to capture the whole of mathematics, but powerful enough to contain just about anything that human mathematicians are interested in.
You can never show conclusively that it is consistent, but you believe it is.
So that's what mathematicians do, that's the basic idea of the mathematical systems such as ZF, ZFC, etc.
By Godel's incompleteness theorem, any powerful enough system of axioms can never capture all of maths.
If you've described it clearly enough so that it's totally clear how theorems are proved - then by his ingenious methods, that leaves it open to processes of adding new axioms to the system, which a mathematician can see must be true, but which are not included in your original list of axioms (it also leaves you open to the possibility of adding in the negations of those new axioms).
But you can then expand your axiom systems as needed in creative ways if you need to go beyond their limitations.
So - it means maths has to be creative, and never ending. And we can never know for sure that it is consistent once it reaches a certain level of complexity and power.
But doesn't have to be inconsistent.
Perhaps same is true of the universe? I don't know.
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
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