As the title say, are the formal sciences (mathematics, logic, etc) fundamentally rationalistic while the natural ones (chemistry and physics) fundamentally empiricist?

Physics, Chemistry, and Biology could never ever have come about without data from the world. There's no way to deduce the celestial motion of the planets without first observing them, and human reason doesn't have (for all we know) the power to rationalize that a body with 2 protons and 2 neutrons (helium) will be more stable than a body with 8 protons and 8 neutrons (oxygen). So, all in all, natural sciences depend on observing the things of the world.

Meanwhile, in Mathematics and Logic, that doesn't follow — in my view at least. The Identity Principle is categoric (A is A and B is B, both are distinct entities as long as A ≠ B), and a square will always, regardless of the name given to it, have 4 sides and 4 edges. These two facts don't depend on collected data in order to be asserted.

Thus, are the formal sciences of the same philosophical vein of Descartes and Leibniz, and the natural sciences of the same philosophical vein of Locke and Hume? Has this correlation ever been done by a prominent thinker?

Edit: I am aware of mathematics being the language of the hard sciences. The question is more related to by what method each of these sciences (formal and natural) expand their knowledge.

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    Natural Sciences build rational models based on empirical data. The rational models, once well developed enough form into your Formal Sciences (as it seems you have described it). You could never deduce the motion of a planet without knowing they existed in the first place. Mathematics couldn't explain gravity until an apple fell down. So you could say they are all based off Empirical data at some point
    – Shadowzee
    Jul 25, 2019 at 2:23
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    One can be a rationalist about natural sciences (Descartes, Husserl), or an empiricist about formal ones (Mill, Quine), so no. It is one's epistemology that makes the rationalist/empiricist divide, not the subject matter. The best one can say is that natural sciences are easier for empiricists to handle, and formal ones are easier for rationalists, some explaining needs to be done for the "other" type. But it has been done at length by both: e.g. Husserl has given an aprioristic account of studying nature, and Quine an empiricist one of doing math.
    – Conifold
    Jul 25, 2019 at 5:41
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    @Conifold I quite like your comment, maybe it could become an answer.
    – user31740
    Jul 26, 2019 at 17:11
  • I would like to offer the notion that rationalism is empiricist. We know the contents of our minds because we have searched them, and talked to other people about the contents of their minds. That is an empirical study. Math is a branch of psychology. In particular it investigates a certain feeling of 'intuitiveness' and how those things which appeal to it do or do not fit together with one another or accord with experience. Apr 21, 2020 at 19:02
  • @William. The only alternative is to believe in fairies. If Math studies something, it is human reactions to patterns. Studying human reactions is psychology. Apr 22, 2020 at 13:52

3 Answers 3


Simple (and rational!) answer is yes

Empirical answer like all empirical data – ostriches and penguins are birds but don't fly etc – is more messy. eg Imre Lakatos showed how surprisingly fallibilistic a historical trajectory math actually takes. And the greatest physicists – Einstein, Newton – have a very strong rationalistic streak : witness Einstein's "God doesn't play dice" in which Einstein effectively says he would prefer his own rationalistic intuition over and above the fact that empirical data was contradicting it. The same intuitions that arguably produced among the most significant science of the twentieth century.

Historical note

Rationalism historically traces to Plato; empiricism to Aristotle. And most "rational" philosophers of recent times would see Plato – heavenly world of Forms etc – as way too top-heavy; "mystical" if one wants to be more derisive.

To paraphrase Voltaire's:

If God did not exist, it would be necessary to invent Him

Likewise also here

The rational – Platonic – world must be assumed real even when that assumption is non-rational.

Empiricism as a corrector of rationalistic excess is fine. As director and definer of science it's nonsense. On its own, mechanical churning of data will only produce machine-learning never the laws of science.

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    As an offside comment, Einstein's quote "God doesn't play dice" refers to his denial of the now-dominant Copenhagen's interpretation of Quantum Mechanics that establishes randomness as an essential element of reality, rather than reason vs empiria debate; Einstein was a determinist, after all. See: en.wikipedia.org/wiki/…
    – user31740
    Jul 25, 2019 at 2:24
  • Not sure what your point is @William. Your links says: As a young man (Einstein) preferred books whose content concerned a whole world view. This is the hallmark of the rational dream: To deduce the larger (at extreme the whole world) from a few principles.
    – Rushi
    Jul 26, 2019 at 7:27
  • My link isn't only the preface for the category "Philosophical Belief", but what comes after, mainly the subcategory "Free will", which explain exactly what the 'god doesn't place dice' concerns. Regarding the a whole world view, I don't think he meant anything deeper than "ideas about reality" — Einstein used metaphors for 'reality' often, such as "God" and the very common "world", where it doesn't actually means the deity or the planet, but nature —, and there's nothing about the quote that suggests this sort of redutionism ("the macro can be reduced to few principles") you imply. @Rusi
    – user31740
    Jul 26, 2019 at 17:08
  • In the end, the 'God doesn't play dice' is just a snarky punch at the Copenhagen interpretation and nothing beyond that afaik; in the very same way as the Schrodinger's Cat thought experiment.
    – user31740
    Jul 26, 2019 at 17:13

Say I squat down and draw figures in the sand with a stick, one with three sides and one with four sides. That's a triangle and a square. They are empirical objects, perfectly visible to everyone present. If someone asks me which is the triangle and which is the square, I point out the sides with my stick: '1, 2, 3', and '1, 2, 3, 4'. Those are numbers, also empirical, and perfectly visible to everyone. We might abstract from the first to a generalized concept of form (the Platonic ideal of a triangle or a square) or from the second to a generalized concept of number (number independent of any particular object being enumerated), but we abstract from physical observations to intuit basic laws of physics as well, so that's not really an issue.

We must know how to count beans before we can do calculus. If we can't count beans, we can't count anything, and math is dead in the water.

The rationalist/empiricist divide is higher-level dispute, in which 18th century thinkers split over methodological issues. Those on the European continent thought that rational introspection was an important tool for getting at some of the basic questions of human life: the nature of 'being' and 'humanness', matters of ethics, problems of sense perception. English thinkers, by contrast, were wary of introspection — thinking of it as loose and subjective, and seeing it as an avenue for religious ideation to inject itself into scientific discourse — and held out for a more strict form of putatively 'objective' observation. Each approach has its strengths and weaknesses, neither is entirely right, but the dispute is deep-seated and (at times) acrimonious. Mathematics is a touchy subject, because empiricists require mathematics for empirical measurement, but (as Analytic Philosophy found out to its dismay), mathematics calls for certain kinds of introspective reasoning.


The difference between rationalism and empiricism is in the method of proof.

Empiricism requires, by the conventional definition and usual understanding of the term, that the proof of your claim be an observation of the material world.

Rationalism requires instead that the proof of your claim be an observation of your own mind.

The widespread contention that this distinction is fundamental is spurious, however. Empirical sciences wouldn't exist as we know them had they to be justified only by observations of the material world. Empirical sciences require from scientists that they first observe their own mind since the percepts they have relative to the material world are all entirely mental events.

And of course you only need to read Descartes's first pages leading to the Cogito to be convinced that understanding the idea of it requires understanding his confrontation of his observation of his own mind, through introspection, and his observation of the material world, through his senses (irrespective of whether any of these things exist as such).

The difference is real but more a matter a degree than of a black-and-white distinction. Science relies on a large extent of what scientists themselves call "thought experiment".

One of the first scientific discovery, and one which is well-known the world over, is Archimedes' principle. The principle states that water exerts an upward force upon any body partially or fully immersed in it and that this force is equal to the weight of the water displaced by the body. Archimedes didn't discover this principle as it is formulated now, but he realised how he could prove whether a crown is of impure gold by immersing it into water. His idea was to measure the volume of the water displaced by the crown as a measure of the volume of the crown. Archimedes is supposed to have shouted "Eureka" while having a bath and presumably observing the level of the water inside the bathtub go up as a result of immersing himself. However, anyone having a bath can experience the Archimedes force as exerted on their own body by water. Swimming certainly seems a lot easier for it. We feel the force. It is an empirical fact. Yet, understanding what is the cause of the force requires an operation of the mind, something entirely rationalistic in essence.

Still, the ultimate proof in the empirical sciences can only be an observation of the material world. If physics, the epitome of empirical sciences, evolved to the point where such a proof could no longer be obtained, it would be perceived more as a rationalist discipline, as seems indeed to be the current status of String Theory.

It is also possible to consider our observation of our own mind (introspection) to be fundamentally an empirical activity.

There is indeed no fundamental difference between observing the material world and observing the pain that you experience whenever you experience pain. The difference is entirely in the fact that two observers will be able to agree that they see a tree or a bird, while only one observer will be able to observe the pain experienced. Yet, we all will experience pain at some point in our lives, as well as such mental phenomena as remembering the past, feeling nauseous, having a logical intuition, having a Eureka moment, etc. And indeed having any idea about the empirical world.

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