I'm trying to make a link between important mathematical breakthroughs in history and the important philosophical schools at the time. I realize that this topic is awfully broad and could be the subject of an entire book, so to simplify it a little I'd like to know which philosophy schools were probably behind great mathematical breakthroughs.

For example, which school of philosophy could have motivated Leibniz and Newton, or Riemann to start thinking about manifolds without relying on an ambient space, or start thinking about spaces with dimensions greater than $3$?

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    $\begingroup$ What makes you think that there must be a school of philosophy "behind" any mathematical breakthrough? $\endgroup$ – Danu Dec 15 '14 at 9:43
  • $\begingroup$ Throughout history art and philosophy have commonly set the course of society, so I don't see why it shouldn't happen the same with math and science. We could even think of mathematics as art, which would make the relation more evident. It may also be the other way around or I may be wrong, an answer along those lines would also be welcome. $\endgroup$ – hjhjhj57 Dec 15 '14 at 9:50
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    $\begingroup$ I'm not sure I agree with your statement, and I certainly don't think it applies as strongly in the case of science and/or mathematics, although one can probably not completely eliminate factors like this. $\endgroup$ – Danu Dec 15 '14 at 10:41
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    $\begingroup$ @Danu Maybe not behind, but there seems to be one around many major ones historically: Plato; Descartes, Leibniz and Newton; Frege, Cantor and Russell. $\endgroup$ – Conifold Dec 17 '14 at 1:59
  • $\begingroup$ @Conifold would you care to elaborate on that in an answer? (Including references about your other comment also?) $\endgroup$ – hjhjhj57 Dec 17 '14 at 6:31

Leibniz and Newton never thought of manifolds outside of ambient space as far as I know, Newton is credited with forging the concept of absolute space, and Leibniz with reducing it to a relational fiction in the style of Aristotle.

The most influential philosopher of science in 19-th century was Kant, specifically his "Copernican revolution" of interpreting time and space as derivative from human perception. This led to their relativization by empiricists, neo-Kantians and logical positivists after non-Euclidean geometries were discovered. Kant directly influenced the thinking of Helmholtz, Mach, Poincare and Einstein. His influence on Riemann was through Herbart, German philosopher who assumed Kant's chair at Königsberg after his death, and "was a brilliant clarifier and interpreter who sharpened several lines of thought in Kant and Fichte, making their ideas more acceptable to scientists". Other than Gauss, Riemann's mentor, Herbart is the only person mentioned by name as an influence in Riemann's famous 1854 lecture "On the Hypotheses which lie at the Bases of Geometry".

Another mathematician who played a major role in developing ideas about multiple dimensions, and was influenced by Kant through Herbart, was Hermann Grassman, who showed that "once geometry is put into the algebraic form he advocated, the number three has no privileged role as the number of spatial dimensions". His main work Ausdehnungslehre is a cross between a mathematical work and a philosophical treatise about the nature of space, which made it very hard to understand, and delayed its appreciation. But Riemann's lecture is written in the same semi-philosophical style, although it is much more readable, and there are some textual similarities between the two. Grassmann sent a copy of Ausdehnungslehre to Gauss in 1844, and there are some reasons to believe that Riemann also read Grassmann's 1853 paper, which was part of exchange with Helmholtz on the theory of vision.

Kant himself wrote in 1747: "The ground of the threefold dimension of space is still unknown... property of threefold dimension... is arbitrary, God could have chosen another... law [from which] extension with other properties and dimensions would have arisen. A science of all these possible kinds of space would undoubtedly be the highest enterprise which a finite understanding could undertake in the field of geometry".

Although Kant later endorsed Euclidean geometry as a priori true, Helmholtz and Poincare found that it could easily accomodate Riemannian perspective by declaring that spatial intuitions are only precise enough for locally Euclidean assertions. They endorsed empiricist and conventionalist views of geometry respectively, both of which influenced Einstein. The same goes for Mach's ideas about inertia and relativity of motion. As another example, Bohr's Copenhagen interpretation of quantum mechanics has Kantian flavor, and adopts conceptual schemes of logical positivism and similar philosophies.

  • $\begingroup$ Great answer! About your first paragraph: I meant Riemann! He's the one that (as far as I know) dropped the idea of an ambient space. $\endgroup$ – hjhjhj57 Dec 17 '14 at 22:55

One has to decide first what exactly we call philosophy. The meaning of this term substantially changed with time. There was a time when all people doing mathematics and sciences could be called philosophers, and what they were doing could be called philosophy. (That the scientific degree in physics and mathematics still called PhD in many countries, is a reminder of that time).

Evidence shows that this practice only ended in the early 19-th century.

The modern meaning of "philosophy" can be roughly defined as "speculations on those matters which are not covered by exact science". So with this definition, the area of philosophy shrinks with time.

On my opinion, philosophy ends just where the science begins. For example, physics was a part of philosophy until Descartes (including him, but not including Galileo), but since the time of Newton, it is firmly science, not philosophy. After that transition, philosophy either stops discussing the subject, or continues, in which case its influence is usually harmful.

Examples: Aristotle's philosophy which Galileo and Kepler had to fight. Attacks of philosophers like E. During on Riemann, non-Euclidean geometry and n>3 dimensional spaces. Attacks of philosophers like E. Mach on atomic theory and statistical mechanics, and so on.

On the positive side I can mention Bolzano, Frege and Russell, who were probably the last mathematicians who were also called philosophers. These were the last examples in mathematics, I guess. (In physics, Einstein was apparently somewhat influenced by philosophy of Hume and perhaps Mach).

Mathematics actually separated from philosophy earlier than other sciences: this happened probably at the time of Plato. His contemporaries like Eudoxos and Theaetetus were already mathematicians, not philosophers, when they wrote about mathematics.

Another matter is that sometimes mathematicians and physicists use the word philosophy for the kind of speculations in their sciences which are not rigorous. Examples: Complementarity Principle (Bohr), Bloch's Principle, Continuity Principle (in mathematics), etc., or to take an older example, that "the Nature abhors the vacuum". But this is different from philosophy done by professional philosophers.

  • $\begingroup$ This is a great answer, but I get the feeling that it doesn't answer my question completely (even though everything you mention is very relevant). Maybe the problem is the question itself, I need to think about it. As @Danu pointed out in the comments, mathematics may be more independent from philosophy than I originally thought. Would you say mathematic's creative force is as essential as philosophy's or art's? $\endgroup$ – hjhjhj57 Dec 15 '14 at 19:50
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    $\begingroup$ Eudoxus (correct Latin spelling) or Eudoxos (correct Greek spelling) wrote also famously about ethics. I do not see why you think he was "already" a mathematician and not a philosopher. $\endgroup$ – fdb Dec 15 '14 at 19:58
  • $\begingroup$ @fdb: thanks for the spelling correction. I mean that Eudoxos was a mathematician when we wrote about mathematics. Mathematicians may have various hobbies and can also do anything else, including alchemistry and theology. $\endgroup$ – Alexandre Eremenko Dec 16 '14 at 0:35
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    $\begingroup$ Mach was actually a philosophizing physicist, as was Einstein, and his philosophy directly influenced his physics. The same can be said of Gödel in mathematics, Grassmann's philosophy is hard to separate from his mathematics. "Philosophy ends just where the science begins" isn't chronological, they remix every time major conceptual difficulties arise in science. $\endgroup$ – Conifold Dec 17 '14 at 0:50
  • $\begingroup$ @Conifold: I agree with your last statement. $\endgroup$ – Alexandre Eremenko Dec 17 '14 at 3:45

Leibniz was a very famous philosopher. So surely you can say that his mathematical discoveries were made against the background of his own “school”. Plato reportedly had the words “Let no one ignorant of geometry enter here” inscribed over the threshold of his school in Athens. The great mathematician Eudoxus of Cnidus (who worked out the first ever viable mathematical model for planetary movements) was a pupil of Plato’s.


Mathematics is made richer with a diverse set of philosophical opinions, but the paradigm that you think might be "golden" for mathematics changes over time, especially in the partly dialectic process of mathematics, which is constantly synthesizing in unknown ways!

Might there be a "Zeitgeist" or spirit of mathematics then? Just a little joke.....

Considering mathematics, and mathematics alone, Platonism might be the most defendable position available for the power it has describing the realism of mathematics. Then, that just defends realism, not how mathematics is actually done in practice.


The question being rather vague I will propose another answer: The French school. Descartes in his algebraic approach was well aware that there was no necessary connection between powers and dimensions, and values greater than 3 were not essentially different. After Newton the understanding that rational or analytic mechanics is geometry + time became rather common. At the end of the 18th. c. Lagrange states:

“La mécanique peut être regardée comme une géométrie à quatre dimensions et l'analyse mécanique comme une extension de l'analyse géométrique /Mechanics can be considered to be a geometry with 4 dimensions/

He was of course aware of the formalism behind inverse functions and derivatives,or more generally the symmetry x(t)<->t(x). However sometime earlier d'Alembert has written:

on pourrait cependant regarder la durée comme une quatrième dimension, et que le produit du temps par la solidité serait en quelque manière un produit de quatre dimensions /one could consider duration as a 4th dimension/

and rather clearly he is indebted to Descartes. Both quotes suggest to say that the French Encyclopedists, who were generally materialistic and mechanistic philosophers, offered the first views of higher dimensional thinking.

Interestingly it was an opponent to Descartes, the Cambridge Platonist Henry More who tried to solve the mind-body problem in his Enchiridion Metaphysicum by proposing a 4th dimension to which only pure spirits have access. His idea was never really forgotten and during the 19th. c. when spiritism was in fashion it is frequently mentioned and one can even find a joking allusion to it in Einstein’s popular writings.

(Refs to this post are to be found in Meyerson’s La Deduction relativiste)


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