Status Updates From Philosophy of Mathematics a...

Many problems come from giving a reality status to space that it does not possess, that involve over applying mathematical models beyond their correct boundaries.

Also on individuals, he mentions how something is not an individual merely through being named but by having a unique description or identifying process, akin, and probably based on Russell's theory of descriptions approach, And relating to space, we see that points in space are not individuals, adding to their secondary "reality" status, while numbers, at least prime numbers I would say, are true individuals.

Some very interesting references to Leibniz in these first couple of sections regarding definitions of space, the ideality of space and how this is connected to realising that space is not a property or predicate of things it is merely an idealised abstraction from applying equivalence relations universally.

A proposition such as 'the rose is red' is no longer subordinated to the scheme' x is red,' having one blank, x, but to the more general one ' x has the property X,' ... (The grotesque confusion of the copula with existence and with equality is one of the saddest indications of the dependence of philosophical speculation on accidental linguistic forms.)

Gödel's construction is closely connected to the logical paradoxes that had provoked so much discussion among the ancient philosophers. The Socratics reveled in paradoxes of this kind. Aristotle dedicates to them a whole book, De Sophisticis Elenchis. The medieval scholastic development culminated in Paulus Venetus (d. 1428). Most modern philosophers, e.g. C. Prantl, dismissed them contemptuously.

Our judgement [of how to reconstruct past events] thus proceeds as if the system with which we are dealing had been created before our time. The word 'creation' suggests a metaphysical or even theological interpretation, but this should not prevent us from recognizing the state of affairs which is most aptly expressed by this word.

The composition of the material world out of one or a few units, existing in a huge number of completely alike specimens, must surely be looked upon as one of the most fundamental features in the nature of the universe, and one that is most profoundly in need of interpretation.

It transpired that for the various chemical elements the atomic masses are far from proportional to the atomic volumes. This shattered the basic conception of one matter, the conception of a homogeneous dough of substance out of which the Creator, with the help of a set of baking moulds, at the beginning of time had carved the little atom cakes, and had then given them absolute rigidity and sent them off into space.

For Galileo, Huyghens and Newton, the deductive part plays a much greater role than in modern times. Galileo is no less proud of the "abundance of theorems which flow from a single principle" than of the discovery of this principle itself.

[Written in 1926]

It must be asked in connection with n-dimensional geometry, what inner reasons there are for the distinction of n=3 realized by actual space. Aristotle gave several answers to this, which still move in the sphere of mythical thought. The solution which Galileo proposes is merely a clearer formulation of the problem. The best chances for success seem to me to lie in theoretical physics construction.

Hume attempted to uphold with inexorable consistency the viewpoint that the given is the whole of reality. Since it became apparent through him that this viewpoint fails completely in the explanation of those cognitive positions which play a basic role in everyday life and in science, he was indeed the first to reveal the problem of reality in its full difficulty.

It is an appealing interpretation of A. Speiser's that Dante, as in Einstein's theory, assumes the real space of creation to be closed rather than bounded. The radii emanating from the center of the Earth, the seat of Satan, converge towards an opposite pole, the center of divine force. Compare the Paradiso, beginning with the 28th Canto. Einstein's space, of course, lacks a distinguished pair of opposite poles.

The supreme act of redemption by the Son of God, crucifixion and resurrection, is no longer the unique pivot of world history but the hurried small-town performance of a road show repeated from star to star - this blasphemy displays perhaps in the most pregnant manner the religiously precarious aspects of a theory which dislodges the Earth from the center of the world. (Bruno had to pay for it at the stake.)

In the projective color plane, the pure spectral colors lie on a curve whose extremities come very close together and are connected by the color purple. Epistemologically it is not without interest that in addition to ordinary space there exists quite another domain of intuitively given entities, namely the colors, which forms a continuum capable of geometric treatment.

If in summing up a brief phrase is called for that characterizes the life center of mathematics, one might say: mathematics is the science of the infinite. It was the great achievement of the Greeks to have made the tension between the finite and the infinite fruitful for the analysis of reality.