From Reuben Hersh's book "What is Mathematics Really".

Any proof has a starting point. So a mathematician must start with some

undefined terms, and some unproved statements. These are "assumptions" or

"axioms." In geometry we have undefined terms "point" and "line" and the

axiom "Through any two distinct points passes exactly one straight line." The

formalist points out that the logical import of this statement doesn't depend on

the mental picture we associate with it. Nothing keeps us from using other

words—"Any two distinct bleeps ook exactly one bloop." If we give interpretations

to the terms bleep, ook, and bloop, or the terms point, pass, and line, the

axioms may become true or false. To pure mathematics, any such interpretation

is irrelevant. It's concerned only with logical deductions from them.

Results deduced in this way are called theorems. You can't say a theorem is

true, any more than you can say an axiom is true. As a statement in pure mathematics,

it's neither true nor false, since it talks about undefined terms. All mathematics

can say is whether the theorem follows logically from the axioms.

Mathematical theorems have no content; they're not about anything. On the

other hand, they're absolutely free of doubt or error, because a rigorous proof

has no gaps or loopholes.

Any proof has a starting point. So a mathematician must start with some

undefined terms, and some unproved statements. These are "assumptions" or

"axioms." In geometry we have undefined terms "point" and "line" and the

axiom "Through any two distinct points passes exactly one straight line." The

formalist points out that the logical import of this statement doesn't depend on

the mental picture we associate with it. Nothing keeps us from using other

words—"Any two distinct bleeps ook exactly one bloop." If we give interpretations

to the terms bleep, ook, and bloop, or the terms point, pass, and line, the

axioms may become true or false. To pure mathematics, any such interpretation

is irrelevant. It's concerned only with logical deductions from them.

Results deduced in this way are called theorems. You can't say a theorem is

true, any more than you can say an axiom is true. As a statement in pure mathematics,

it's neither true nor false, since it talks about undefined terms. All mathematics

can say is whether the theorem follows logically from the axioms.

Mathematical theorems have no content; they're not about anything. On the

other hand, they're absolutely free of doubt or error, because a rigorous proof

has no gaps or loopholes.

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