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Kant's proof of the existence of synthetic apriori knowledge was a response to Hume's fork and his views the problem of induction.

Given the relevance of these two concepts to the philosophy of science in general and to the demarcation problem in particular, can it be said that Kant's synthetic apriori had any influence on the philosophy of science?

Does it have any implications for the demarcation problem?

Given Kant's pervasiveness in philosophy in general, has he had any impact on philosophy of science, not just due to the synthetic apriori result?

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There is an unbroken chain of tradition from Kant to all major currents in the philosophy of science. As for demarcation, Kant's standard was far stricter than even Popper's. For example, he called chemistry "systematic art or experimental doctrine but not a proper science", and infamously opined that empirical psychology will never become a science because introspection data is too garbled and ephemeral. Only that which could structure experience according to synthetic a priori principles in a mathematical form (i.e. mathematical physics) deserved the name of (natural) science. He somewhat softened his stance in late years, willing to find a place for chemistry as a kind-of-science in light of Lavoisier's work however. I still suspect that not even modern biology, let alone soft sciences, would pass Kant's muster.

The assimilation of Kant by scientists proceeded through Herbart, a talented popularizer who succeeded him at Königsberg, and to a lesser extent Fichte, who spelled out what Kant's "transcendental method" was. Dissatisfied with speculations of German idealism Helmholtz issued the "back to Kant" call in his 1855 lecture. The original interest was in Kant's theory of sensibility in relation to the emerging physiology of senses, especially vision and touch (early studies of vision motivated Kant's model of sensibility, productive imagination and understanding), and the related subject of empirical geometry (Riemann names Gauss and Herbart as the only influences in his famous lecture, see Which school of philosophy motivated thinking about spaces of higher dimension?). The approach was to discard idealist elements in Kant and relax his synthetic a priori to make room for more empirical input, e.g. a priori did not strictly dictate Euclidean geometry, only some locally Euclidean one, to be determined empirically. Poincare went further in accepting the creative role of mental faculties in science, and postulated that empirical theories only reflect certain structures of experience (Kant would say those brought under the categories of understanding), e.g. experience only specifies geometry + physics but not each part separately, that is decided by convention. Conventional aspects are determined by pragmatic reasons and may be discarded in new theories. Current structural realists (Worrall, Tegmark) trace their lineage to Poincare.

Professional philosophers of science went back to Kant soon after Helmholtz. Marburg neo-Kantians introduced two approaches that came to dominate the assimilation of Kant: externalization and relativization. Kant tacitly identified scientific process with individual cognition assuming that scientific discovery follows the mental a priori. Around 1870 Cohen, the founder of Marburg school, proposed to get rid of this "psychologism", and explicitly identified knowledge with scientific knowledge, and replaced Kant's things in themselves with an object of knowledge as ideal limit approached through a succession of refinements. Cassirer played a key role in transmitting the neo-Kantian creed to logical positivists. After Einstein's formulation of relativity Reichenbach, an early logical positivist known for coining "context of justification" vs. "context of discovery", explicitly suggested that a priori should be relativized to account for scientific succession, they last longer than the theories they help generate, but even they are updated from time to time. Finally, in Carnap, the logical positivist par excellence, we find Kant fully externalized and relativized, one could even say formalized. Pure reason (single) becomes a linguistic framework (one of many), synthetic a priori become its definitions, axioms and application rules, the duality between sensibility and understanding, and hence appearances and experience, becomes the duality between observation and theoretical languages, and the noumena/phenomena distinction becomes the distinction between external and internal questions, the former being pseudo-questions devoid of meaning within the framework. For all their disagreements, Popper also accepted the key elements of this picture, formalization of scientific theories and the observation/theory distinction. It was a neat picture, until Quine and Kuhn ruined it.

The elimination of sensibility as independent faculty by Cassirer (in response to relativity and other developments in physics) ultimately led to impossibility of maintaining the observation/theory distinction. And along with taking relativization of a priori to the extreme (of discarding them) it paved the way for Quine's holism. Kuhn's culturalization and historicization of philosophy of science somewhat paralleled Hegel's revision of Kant's philosophy with inspiration delivered through Marx. More specifically, through Soviet Marxist Hessen, the Hessen thesis injected social context into Western historiography of science in 1930s. Interestingly enough, Carnap saw affinities between his linguistic frameworks and Kuhn's paradigms, and even published Scientific Revolutions in his book series. It took Kuhn some time but he saw the reciprocal light as well. Moreover, both Quine and Kuhn walked back the more radical formulations of holism and cultural relativism, and Friedman, a contemporary neo-Kantian, showed how their more moderate versions can be accomodated in a scheme of historically evolving paradigms structured by relativized a priori in Dynamics of Reason.

The pursuit of more radical versions led to the postmodernistic pandemonium of 1980s (on both sides of the analytic/continental divide) ending in the grotesque grand finale of Sokal hoax. The story is told in Zammito's Derangement of Epitemes. The resurgence of structural realism and even naive realism (Weinberg) among natural scientists in recent decades is likely a backlash against that anti-scientific postmodernistic exuberance.

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    Great. Marvelously comprehensive answer! Are there any related books you can recommend? In addition to the post-Kantian influence on sciences, I am interested in the Kantian Marxism of the Austro-Marxists and after, which seems to be a marxism better suited to contemporary financialization themes. As a pure hobbyist with limited time, relatively concise works are better for me. – Nelson Alexander Feb 9 '16 at 15:12
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    @Nelson Alexander Zammito's book covers the 1950-2000 period, neo-Kantianism, transition to logical positivism and the emergence of analytic/continental divide are nicely analyzed in Friedman's Parting of the Ways, Cassirer's Problem of Knowledge is considered a classic on 19th century. See also short paper Kant, Herbart and Riemann by Banks philpapers.org/archive/BANKHA.pdf – Conifold Feb 9 '16 at 18:56
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Keynes once quipped that public figures who think they are expressing their original thoughts are usually echoing the words of some dead economist. The same might be said of the dead Kant in respect to science. While his thought provides a comprehensive modern framework for science, most practicing scientists have never read him and large swaths of the philosophy of science ignore or reject what they take to be muddled Kantianism.

Bertrand Russell and E.G. Moore were especially hostile to Kant, convicting him of logical errors and supposing that transcendental idealism rested on a mistaken faith in the inviolability of Euclidean Geometry, which Kant presented as the very model of the "synthetic a priori." These dubious charges stuck all the way through logical positivism and continue in the analytic tradition. Certainly Popper's demarcation criteria seem to reject Kantian approaches, or at least so Popper claimed.

Today, however, it seems that more philosophers of science may be open to Kant. Henry Allison offers detailed refutations of the above calumnies, and Kuhn was explicitly influenced by Kant in his anti-Popperian demarcation by "paradigms." He claimed that reading Kant while studying physics utterly altered his naive realism, though those are not his words.

As to the bigger picture. Bacon first described the emerging rift between the rationalists, "the spiders," who weave webs of theory (coherence theory, we might say) and the new empirical naturalists, "the busy ants," who gather bits of data (correspondence theory, roughly). The rift grew into Leibnizian mathematical rationalism and Humean skepticism. It was Kant's great project to merge and mutually limit the two on a firm metaphysical basis, in part to secure the basis of Newtonian physics.

His unique amalgam of coherence and correspondence theory is in many ways, and intentionally, a kind of philosophical version of the hypothetic-deductive framework of science, an expansion of knowledge by rational (conceptual) methods and experimental (intuitional) confirmation. And indeed science does undertake continuous active synthesis based on a priori assumptions of necessity and universality. Even the "experiment" is somewhat Kantian, in that it is hardly passive reception of sense data... the experiential confirmation in "common sense" is quite artificial and actively constructed.

Most working scientists tend to believe they are following analytical rules and discovering "correspondences" with hard "reality." So they look askance at Kantian idealism, regarding it as akin to Berkeleyan empirical idealism or some sort of structuralism. Such assurance was, of course, shaken by statistical mechanics in thermodynamics and then quantum mechanics. But such "shaking" may only register with a few...how many physicists, after all, have time to read Kant? Yet when pressed on theoretical issues many physicists will admit they are constructing models and cannot speak about the "ultimate reality," mere speculative metaphysics, unaware that they are defaulting to a near Kantian position.

Meanwhile, cosmology today seems to be weaving "theoretical webs" well outside any Kantian remit, moving far beyond experimental range and tumbling headlong into the Antinomies. In CPR B511, for example, we see hints of Copenhagen in "you never come face to face with anything unconditioned...." or "...neither a simple appearance [i.e., final particle] nor an infinite composition [i.e., material universe] can ever come before you." One might say Kant attempted to work out a physics that included the observer.

Of course, this is very general and it is easy to contrive this sort of cozy compatibility. I don't know enough about Kant yet to know where he might be in serious conflict with scientific practices, especially in theory of evolution... or perhaps in the evolution and "selection" of theories, where Hegel's criticism of Kant begins. I hope other will offer more specific references to Kant in current philosophy of science, since I'd like to know as well.

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    "Meanwhile, cosmology today seems to be weaving "theoretical webs" well outside any Kantian remit, moving far beyond experimental range and tumbling headlong into the Antinomies." -- This could be the topic of a separate discussion. I like the way you put it. – Alexander S King Feb 9 '16 at 18:07
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First let me recall the view which is denoted “Hume’s fork” (see http://plato.stanford.edu/entries/hume/):

Propositions concerning relations of ideas are intuitively or demonstratively certain. They are known a priori—discoverable independently of experience by “the mere operation of thought”, so their truth doesn't depend on anything actually existing (EHU 4.1.1/25). That the interior angles of a Euclidean triangle sum to 180 degrees is true whether or not there are any Euclidean triangles to be found in nature. Denying that proposition is a contradiction, just as it is contradictory to say that 8×7=57.

In sharp contrast, the truth of propositions concerning matters of fact depends on the way the world is. Their contraries are always possible, their denials never imply contradictions, and they can't be established by demonstration. Asserting that Miami is north of Boston is false, but not contradictory. We can understand what someone who asserts this is saying, even if we are puzzled about how he could have the facts so wrong. The distinction between relations of ideas and matters of fact is often called “Hume's Fork”, […].

Kant does not refute the distinction made by this classification. Like Hume also Kant accepts mathematical truths as example of “Relations of ideas” and many statements from physics as propositions concerning “matters of fact”. But different from Hume Kant holds that some – synthetic - statements from physics can be known a priori, see Kant’s Metaphysical Foundations of Natural Science.

Your question asks about Kant’s influence on the philosophy of science.

1) Quine in his seminal essay “Two dogmas of empiricism” (1951) confirms:

Kant’s cleavage between analytic and synthetic truths was foreshadowed in Hume’s distinction between relations of ideas and matters of fact […].

Quine names the belief in this distinction the first dogma of empiricism and he refutes it. He links the first dogma with a second dogma (reductionism) and states in paragraph 5:

My countersuggestion […] is that our statements about the external world face the tribunal of sense experience not indivudually but only as a corporate body.

As a consequence it makes no sense for Quine assessing single propositions and classifying them as analytic or synthetic.

2) Broadly speaking, Popper returns to Hume’s view of induction and adds the concept of falsification of scientific propositions. Hence Popper refutes the existence of Kant’s view on the existence of synthetic a priori statements. As a consequence the distinction of analytic and synthetic does not play an important role in Popper’s demarcation of science and metaphysics – I agree with Mauro’s answer.

3) http://plato.stanford.edu/entries/kant-science/ testifies a certain influence of Kant’s thoughts on contemporary philosopyh of science. The passage starts:

Kant's philosophy of science has received attention from several different audiences and for a variety of reasons. It is of interest to contemporary philosophers of science primarily because of the way in which Kant attempts to articulate a philosophical framework that places substantive conditions on our scientific knowledge of the world while still respecting the autonomy and diverse claims of particular sciences.

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