Book I, 4, against Anaxagoras: proposed principles: if the whole is known then so too are the parts. Also: there is no knowledge of the infinite. The real numbers contain a subset of indefinable numbers that are individually unknowable. Therefore either they are not parts of the real numbers or the real numbers are as a whole unknowable.
Book III: no infinite body in space. In some of the proofs Aristotle's show the extreme conceptual importance and interest behind the framework of the apparently naive theory of the four elements. The idea of opposition of a quality is intimately connected to the negative numbers, the balancing out of opposites. Thus both dry-wetness and hot-coldness represent independent genera which can be parametrized (or scaled) by $\mathbb{R}$, where for instance wetness and coldness are negative. The concept of electric charge of modern physics is here. In the proof Aristotle basically argues that the total charge in the universe must add up to zero. Now what is most interesting about the theory of the four elements is that we have matter endowed simultaneously with two independent kinds of charges. So to parametrize the field of fundamental qualities we need $\mathbb{R}^2$ or more suggestively $\mathbb{C}$. Thus $e^{i\theta t}$ could represent the circulation and transformation of the elements. The fact that fundamental qualities can either be scalar (positive reals) or vectors (one or higher-dimensional) (i.e. combinations of continuous genera having opposites) goes back to Aristotle.
Book VI: no finite movement in an infinite time. The magnitude of a movement is shown by Aristotle to be closed and bounded (measured by intervals), hence compact. Infinite time can only mean a semi-open set $[0,\infty)$. If motion is continuous then the result follows from the fact that there is no continuous bijective function (homeomorphism) $f: [0, \infty) \rightarrow I$ where $I$ is some closed interval (all with the topology induced by the standard topology). To us this follows immediately from the fact that $f^{-1}$ must take the compact set $I$ into a compact set. But $[0,\infty)$ is not compact. Aristotle's argument involves dividing $I$ into subintervals $I_i$ and observing that the $f^{-1}I_i$ have to be finite and hence $f^{-1} I = f^{-1}(\bigcup_i I_i) = \bigcup f^{-1}I_0 = [0,\infty)$ would have to be finite (for finite times finite is finite) which is a contradiction. This argument works is we observe that $f^{-1}$ is assumed to be a continuous function $g$. However Aristotle's arguments seems to rest rather on the monotonicity of the function implying that $g(I_i)$ must itself be a bounded interval $[0,a)$. Aristotle did not have our concept of 'function', he thought rather in terms of correspondences or relations, perhaps somewhat similar to adjunct pairs. Thus there is a bilateral correspondence in the Physics between time, magnitude, motion and even the extension of the moving body itself. A division of one of these quantities must correspond to a division in the others (but perhaps not the moving body itself).
Book VI: no first time for started moving. Aristotle constructs a filtration $U_1 \supset U_2...\supset U_i \supset...$ of shrinking semi-intervals all containing the starting point $t_0$. This is $dt_0$.
If at time $t$ a body $M$ is moving then it has moved before and it has been moving before. Moving times are an open set. $t$ is both a limit and there is $t' < t$ such that $M$ is moving at $t'$.
No comments:
Post a Comment