Do you agree that an axiom is not merely a conjecture that is accepted by the vast majority of educated people?

[PseuTonym]

Obviously, if a statement is actually false, regardless of how plausible it appears to be, then it is not an axiom. It is possible for the vast majority of educated people to agree on some question, and for the statement that is agreed upon to actually be false. However, there is something more that needs to be said.

**A statement might not be an axiom, even if there is universal agreement that it is an axiom, and even if the statement is actually true.
**
For example, there is a statement that was a conjecture and that was known as “Fermat’s Last Theorem.” It is now considered to be an actual theorem, but the simplest known proof of it relies upon assumptions that were not merely awaiting acceptance by the mathematical community when Fermat formulated the conjecture. It relies upon assumptions that had not yet been thought of or formulated by anybody. On the basis of those assumptions, Andrew Wiles created a proof.

Fermat’s Last Theorem is not an axiom. Even if all mathematicians and all schools and all governments in the world assigned the label “axiom” to it, it would not be an axiom. However, if somebody discovers positive whole numbers x, y, z, and n with n larger than 2 and (x to the power of n)+(y to the power of n) = (z to the power of n), and announces the discovery for verification around the world, then Fermat’s Last Theorem will – for very good reasons – no longer be considered a theorem.

 

[grannymh]

Obviously, if a statement is actually false, regardless of how plausible it appears to be, then it is not an axiom. It is possible for the vast majority of educated people to agree on some question, and for the statement that is agreed upon to actually be false. However, there is something more that needs to be said.

**A statement might not be an axiom, even if there is universal agreement that it is an axiom, and even if the statement is actually true. **

For example, there is a statement that was a conjecture and that was known as “Fermat’s Last Theorem.” It is now considered to be an actual theorem, but the simplest known proof of it relies upon assumptions that were not merely awaiting acceptance by the mathematical community when Fermat formulated the conjecture. It relies upon assumptions that had not yet been thought of or formulated by anybody. On the basis of those assumptions, Andrew Wiles created a proof.

Fermat’s Last Theorem is not an axiom. Even if all mathematicians and all schools and all governments in the world assigned the label “axiom” to it, it would not be an axiom. However, if somebody discovers positive whole numbers x, y, z, and n with n larger than 2 and (x to the power of n)+(y to the power of n) = (z to the power of n), and announces the discovery for verification around the world, then Fermat’s Last Theorem will – for very good reasons – no longer be considered a theorem.

I would like to talk about axioms, but “Fermat’s Last Theorem” is beyond me. When I think about axioms, I see them as being truthful.

 

[PseuTonym]

I would like to talk about axioms, but “Fermat’s Last Theorem” is beyond me.

Perhaps some examples would help. What the Fermat conjecture known as “Fermat’s Last Theorem” asserts is rather simple, and many students in grade 6 or 7 can understand it. I doubt that it is beyond you. Understanding an alleged proof of the conjecture is another matter.

The reason that I mentioned Fermat’s Last Theorem is that it illustrates the difference between theory and experiment. Fermat’s Last Theorem could be shown to be false by means of a non-controversial and finite computation, provided that the data exists and that somebody discovers it. In that sense, the hypothesis that Fermat’s Last Theorem is true is an example of what Karl Popper would call a “falsifiable” (and hence “scientific”) claim. Note that “scientific” does not mean “true.” According to Popper, it means that if the hypothesis is false then it can be definitely refuted.

When I think about axioms, I see them as being truthful.

Yes, they ought to be truthful. However, there are now libraries and buildings and college departments of “the mathematical sciences.” There are many people who are no longer willing to consider saying that there are some statements that they have faith in. If they are going to use deductive logic to build on some foundation, then they want to assure themselves and others that the foundation is solid. However, mere naming can never provide such an assurance.

I hope that in future some people will have enough insight to not only formulate new statements that seem to be true, and that could be a foundation for future mathematical deductions. I hope that they are honest enough to avoid claiming that they are building on “scientific fact” or “mathematical axioms.” I hope that in future we see some new mathematical faith that is explicitly described as being a mathematical faith.

 

[Just_Lurking]

In set theory, the axiom of choice (see here) and the axiom of determinacy (see here) are incompatible with each other, in the sense that if one is true, the other must be false. Of course, they could both be false.