ONT Re: Ontology as Math or Metaphysics?
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Note 5
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I see that one is likely to get a false impression by
reading only the half of the Paragraph that MK quoted,
so here is the whole Paragraph restored, where I have
broken the second paragraph of the Paragraph in twain,
for ease of reading and comparison:
| Kant is entirely right in saying that, in drawing those consequences,
| the mathematician uses what, in geometry, is called a "construction",
| or in general a diagram, or visual array of characters or lines. Such a
| construction is formed according to a precept furnished by the hypothesis.
| Being formed, the construction is submitted to the scrutiny of observation,
| and new relations are discovered among its parts, not stated in the precept
| by which it was formed, and are found, by a little mental experimentation,
| to be such that they will always be present in such a construction. Thus,
| the necessary reasoning of mathematics is performed by means of observation
| and experiment, and its necessary character is due simply to the circumstance
| that the subject of this observation and experiment is a diagram of our own
| creation, the conditions of whose being we know all about.
|
| But Kant, owing to the slight development which formal logic
| had received in his time, and especially owing to his total
| ignorance of the logic of relatives, which throws a brilliant
| light upon the whole of logic, fell into error in supposing that
| mathematical and philosophical necessary reasoning are distinguished
| by the circumstance that the former uses constructions. This is not true.
| All necessary reasoning whatsoever proceeds by constructions; and the only
| difference between mathematical and philosophical necessary deductions is
| that the latter are so excessively simple that the construction attracts
| no attention and is overlooked. The construction exists in the simplest
| syllogism in Barbara. Why do the logicians like to state a syllogism by
| writing the major premiss on one line and the minor below it, with letters
| substituted for the subject and predicates? It is merely because the reasoner
| has to notice that relation between the parts of those premisses which such a
| diagram brings into prominence. If the reasoner makes use of syllogistic in
| drawing his conclusion, he has such a diagram or construction in his mind's
| eye, and observes the result of eliminating the middle term. If, however,
| he trusts to his unaided reason, he still uses some kind of a diagram
| which is familiar to him personally.
|
| The true difference between the necessary logic of philosophy and
| mathematics is merely one of degree. It is that, in mathematics, the
| reasoning is frightfully intricate, while the elementary conceptions are
| of the last degree of familiarity; in contrast to philosophy, where the
| reasonings are as simple as they can be, while the elementary conceptions
| are abstruse and hard to get clearly apprehended. But there is another
| much deeper line of demarcation between the two sciences. It is that
| mathematics studies nothing but pure hypotheses, and is the only
| science which never inquires what the actual facts are; while
| philosophy, although it uses no microscopes or other apparatus of
| special observation, is really an experimental science, resting on
| that experience which is common to us all; so that its principal
| reasonings are not mathematically necessary at all, but are only
| necessary in the sense that all the world knows beyond all doubt
| those truths of experience upon which philosophy is founded.
| This is why the mathematician holds the reasoning of the
| metaphysician in supreme contempt, while he himself, when
| he ventures into philosophy, is apt to reason fantastically
| and not solidly, because he does not recognize that he is
| upon ground where elaborate deduction is of no more avail
| than it is in chemistry or biology.
|
| Charles Sanders Peirce, 'Collected Papers', CP 3.560.
| Published in 'Educational Review', pp. 209-216, 1898.
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