9. Conclusion

I have come to the conclusion that Reality has given Humanity a gift that seems to go beyond Reality itself, namely the capability we human individuals have of telling/accepting the one and only story of the one and only Reality which is simply true.

Both Ludwig Wittgenstein, Russell’s student, and Karl Popper, another remarkable contemporary thinker, rejected Bertrand Russell’s “Logic”. But their reasons for that were different, and they famously argued about who was right.

The wonderful fact of the matter, told in this Essay, is that Reality/Nature has given human beings the ability to grasp simple special languages, called sciences, whose claims are both provable, with proofs you and I may agree about, and repeatedly testable, with results that you and I may agree about.

That is a moral view which supports Ludwig Wittgenstein, insofar as the agreements of individuals about what they truthfully claim are expressible in sentences whose meanings we agree about; and it is a moral view which supports Karl Popper, insofar as those agreements are testable by individuals who agree with one another about the meanings of their sentences. In this way, the idea of objective truth acquires an explicitly empirical meaning. A proof of a theorem of some particular science is correct/right for you, if you believe it either because you find it compelling or because you trust others who find it compelling; and an experimental test of the theorem, as a prediction, succeeds if you personally perform that experiment, or you believe others who say they have performed that experiment.

I want to finish this Essay with the outline of a challenging proof that you, the reader, may be able to work though and find compelling, as I do. It invokes the special algebra of vectors that was not available to Euclid. I hope that this proof will persuade some readers that the mathematics of Euclidean geometry is both simple and useful.

Varignon’s Theorem is a remarkable theorem of Euclidean geometry that was not mentioned in Euclid’s great book. Like the Theorem of Pythagoras, Varignon’s Theorem has both an empirical aspect and a reason/logic aspect and both of these are essential.

The empirical aspect of Varignon’s Theoren is easy to state but surprising. Please carefully try this Euclidean geometry experiment:

1 use a ruler to draw a quadrilateral (a four-sided figure with four vertices);

2 join the mid-points of its four sides.

Does the four-sided figure you carefully drew look somehow special? In fact it is the special kind of quadrilateral called a parallelogram, with opposite sides parallel and equal in length.

That is Varignon’s Theorem ––– an empirical fact about Reality, which can repeatedly be tested by seriously following steps 1 and 2.

This theorem/prediction has many proofs. The proof I will outline, is transparently about meanings. It will turn out to involve equations, each one claiming that the phrase on the right-side of the = sign means the same to me/you as does the phrase on the left-side of the = sign.

In order to talk about your drawing on a flat sheet, let us give the names A,B,C,D,P,Q,R,S to 8 particular points of interest. To say that P is the mid-point, of the line segment from A to B, let us write simply P=1/2(A+B). That is just a matter of translating from English to a simple mathematical language which supports basic calculation. Similarly, let us write Q=1/2(B+C), R=1/2(C+D) and S=1/2(D+A).

Also, let us give the name B-A to the vector from A to B. The important notion of a vector was introduced in the early nineteenth century by Hermann Grassmann. Briefly, a vector has both a length and a direction, but no specific position. We would like to show that P,Q,R,S are the vertices of a parallelogram; in the language of vectors, that comes down to showing that Q-P=R-S.

Here now is the complete proof:






The bottom line here is that, Aristotle and Russell and their many followers who, still today, call themselves  "Philosophers of Mathematics" are  incorrect/wrong about mathematics. Mathematics is not the manipulation of meaningless expressions that mindless computers can do. Rather it is the systematic manipulation of meaningful expressions that can be done by the human beings who invent/understand those meanings.