PM5399: True Mathematics

This is an account of TRUE MATHEMATICS
Let us begin with TRUE GEOMETRY

The Geometricans say:   a point has no depth or width or height; it is
without measure. But the truth says: there is naught without measure,
naught without breadth or width or height. Whatever is without these,
is not; therefore indeed, there are naught such as these points of
which they speak. The Geometricans speak of position without
magnitude: but there is no position without magnitude, nor magnitude
without position.

The Geometricans say: a line has infinite length, but zero is its
width. But the truth says: There is naught of zero depth or width or
height; nor is there anything of infinite width or depth or height.

The Geometricans say: Two circles touch at a single point; but the
truth says: There being no points, they cannot so touch – two circles
either do not touch, or else they have a non-zero area of
intersection.

The Mathematicians say: There exists infinity, and an infinity of
infinities even. But the truth says: There is no infinity, neither any
infinities – no such things are. The Mathematicians say: There is an
infinity of whole numbers. But the truth says: There are only finitely
many whole numbers, there being no such thing as infinity. The
Mathematicians say: There is no greatest whole number. The truth says:
There is a finite greatest whole number, such that no whole number is
greater. And the Mathematicians will object: But what is one more than
that number – surely that is also a whole number? But the truth
responds: That is a number without a successor; that is a number for
which the addition of one, or the multiplication by two, yields no
result – for an operation never executed has not any result. And the
Mathematicians shall ask: Pray tell, then, what is its value? The
truth responds: We know that it is, but we know not what it is; dare
we say, we shall never know what it is – or if we ever shall, we shall
so only when we have given up entirely every project of increasing our
mathematical knowledge. For, if we have not given up, we would surely
ask “What is one greater?”, and in so asking disprove our knowledge.
But, when there is not a soul remaining, living or dead, who seeks
after the increase of mathematical knowledge – until forgetfulness
returns all this knowledge to naught once more – then may at last this
number be revealed to us, when we will no longer seek to find any
number greater there-than.

And the Mathematicians say: there are infinitely many real numbers,
and a greater infinity than the whole numbers. But the truth says:
there are finitely many whole numbers; and finitely rational numbers;
and finitely many suprarational numbers.
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