Philosophy

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April 10, 2026

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April 10, 2026

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"Up to this point we have restricted our attention to space as a mere extension. But space, as understood in common practice, implies considerably more: it represents a three-dimensional Euclidean continuum. When thus particularised, Kant's arguments as to its a priori character are no longer tenable in the light of modern discovery; and we must assume that this special form we credit to space arises entirely from our co-ordination of sense impressions conducted in the simplest way possible. On no account may we consider three-dimensional Euclidean space to be imposed a priori either by sensibility or by the understanding. ...it is generally conceded by scientists that the a priori doctrine of three-dimensional Euclidean space is one of the most pernicious teachings that philosophy has ever attempted to impose upon science. ... These views on space as professed by the greatest scientists are in large measure to be attributed to the discoveries of non-Euclidean geometry supplemented by the investigations of the psychophysicists. ...By the time men are of an age to philosophise, they have been subjected for so many years to beliefs based on inferences from experience, that the beliefs have remained, whereas the inferences, owing to the monotony of their repetition, have become second nature and appear intuitive. ... Were three-dimensional Euclidean space an a priori condition of the understanding, it would have been quite impossible for mathematicians to wend their way through the non-Euclidean hyperspaces of relativity. Neither can three-dimensional space be considered to be imposed by sensibility, since, as PoincarĂŠ tells us, after a certain amount of perseverance, he was aided to a considerable degree by sensibility when investigating the problems of Analysis Situs of four dimensions."

- A priori and a posteriori

• 0 likes• philosophy• phrases•
"In opposition to a Hegelian emanationist epistemology, briefly, Neo-Kantians shared the Kantian dichotomy between reality and concept. Not an emanent derivative of concepts as Hegel posited, reality is irrational and incomprehensible, and the concept, only an abstract construction of our mind. Nor is the concept a matter of will, intuition, and subjective consciousness as Wilhelm Dilthey posited. According to Hermann Cohen, one of the early Neo-Kantians, concept formation is fundamentally a cognitive process, which cannot but be rational as Kant held. If our cognition is logical and all reality exists within cognition, then only a reality that we can comprehend in the form of knowledge is rational — metaphysics is thereby reduced to epistemology, and Being to logic. [...] Occupying the gray area between irrational reality and rational concept, then, its question became twofold for the Neo-Kantians. One is in what way we can understand the irreducibly subjective values held by the historical actors in an objective fashion, and the other, by what criteria we can select a certain historical phenomenon as opposed to another as historically significant subject matter worthy of our attention. In short, the issue was not only the values to be comprehended by the seeker of historical knowledge, but also his/her own values, which are no less subjective. [...] Bridging irrational reality and rational concept in historical science, or overcoming hiatus irrationalis (à la Lask) without recourse to a metaphysics of history still remained a problem as acutely as before. While accepting the broadly neo-Kantian conceptual template as Rickert elaborated it, Weber's methodological writings would turn mostly on this issue."

- Neo-Kantianism

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"It will seem a little paradoxical to ascribe a great importance to observations even in that part of the mathematical sciences which is usually called Pure Mathematics, since the current opinion is that observations are restricted to physical objects that make impression on the senses. As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of numbers. Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations; they will lead us continually to new properties which we shall endeavor to prove afterwards. The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful."

- Inductive reasoning

• 0 likes• mind• philosophy•