Author Topic: Schopenhauer's Philosophy of Mathematics  (Read 10167 times)

0 Members and 1 Guest are viewing this topic.

Holden

  • { ∅, { ∅ } }
  • Posts: 5070
  • Hentrichian Philosophical Pessimist
Re: Schopenhauer's Philosophy of Mathematics
« on: April 29, 2020, 05:58:59 am »
Schopenhauer on Laws of Thought:

Four laws
"The primary laws of thought, or the conditions of the thinkable, are four: – 1. The law of identity [A is A]. 2. The law of contradiction. 3. The law of exclusion; or excluded middle. 4. The law of sufficient reason." (Thomas Hughes, The Ideal Theory of Berkeley and the Real World, Part II, Section XV, Footnote, p. 38)

Arthur Schopenhauer discussed the laws of thought and tried to demonstrate that they are the basis of reason. He listed them in the following way in his On the Fourfold Root of the Principle of Sufficient Reason, §33:

A subject is equal to the sum of its predicates, or a = a.
No predicate can be simultaneously attributed and denied to a subject, or a ≠ ~a.
Of every two contradictorily opposite predicates one must belong to every subject.
Truth is the reference of a judgment to something outside it as its sufficient reason or ground.
Also:

The laws of thought can be most intelligibly expressed thus:

Everything that is, exists.
Nothing can simultaneously be and not be.
Each and every thing either is or is not.
Of everything that is, it can be found why it is.
There would then have to be added only the fact that once for all in logic the question is about what is thought and hence about concepts and not about real things.

— Schopenhauer, Manuscript Remains, Vol. 4, "Pandectae II", §163'
To show that they are the foundation of reason, he gave the following explanation:

Through a reflection, which I might call a self-examination of the faculty of reason, we know that these judgments are the expression of the conditions of all thought and therefore have these as their ground. Thus by making vain attempts to think in opposition to these laws, the faculty of reason recognizes them as the conditions of the possibility of all thought. We then find that it is just as impossible to think in opposition to them as it is to move our limbs in a direction contrary to their joints. If the subject could know itself, we should know those laws immediately, and not first through experiments on objects, that is, representations (mental images).

— Schopenhauer, On the Fourfold Root of the Principle of Sufficient Reason, §33'
Schopenhauer's four laws can be schematically presented in the following manner:

A is A.
A is not not-A.
X is either A or not-A.
If A then B (A implies B).
Two laws
Later, in 1844, Schopenhauer claimed that the four laws of thought could be reduced to two. In the ninth chapter of the second volume of The World as Will and Representation, he wrote:

It seems to me that the doctrine of the laws of thought could be simplified if we were to set up only two, the law of excluded middle and that of sufficient reason. The former thus: "Every predicate can be either confirmed or denied of every subject." Here it is already contained in the "either, or" that both cannot occur simultaneously, and consequently just what is expressed by the laws of identity and contradiction. Thus these would be added as corollaries of that principle which really says that every two concept-spheres must be thought either as united or as separated, but never as both at once; and therefore, even although words are joined together which express the latter, these words assert a process of thought which cannot be carried out. The consciousness of this infeasibility is the feeling of contradiction. The second law of thought, the principle of sufficient reason, would affirm that the above attributing or refuting must be determined by something different from the judgment itself, which may be a (pure or empirical) perception, or merely another judgment. This other and different thing is then called the ground or reason of the judgment. So far as a judgement satisfies the first law of thought, it is thinkable; so far as it satisfies the second, it is true, or at least in the case in which the ground of a judgement is only another judgement it is logically or formally true.

(From Wikipedia)
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.