In recent years the steadily increasing computing capacity of computers has led to a lot of new areas in science. In most cases the computer is used to process huge sets of data which cannot possibly be analyzed by hand. For example:

  • finding proofs in maths and physics by «simply» testing every possibility (brute force)
  • simulations (Monte-Carlo, climate models, …)
  • automated filtering of large bulks of data using prescribed criteria

Naturally, it is not possible to check the results by hand and one has to trust that the result is "accurate". It seems likely that the importance of computers will further increase in the future and that many simulations will be done whose results cannot be compared to real experiments.

What are consequences for the philosophy of science? Is it necessary to develop additional theories for the new techniques or are they not really new and can simply be treated like, e.g., classical experiments?

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    Computations are a priori so there isn't an issue with using a computer to compute the outcome of an experiment. The only issue involves something like "well how do I know the wiring in my computer isn't faulty?" But that isn't an issue with the computation itself, that's an issue with its implementation. I don't think the philosophy of science would have an issue because we know that the math is correct. The bigger questions are "are we using the correct algorithm and data? Is my machine working?" But those aren't the same type of question as "is math a priori and therefore safe to assume?"
    – Not_Here
    Sep 5, 2017 at 16:28
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    Computer assisted proofs do indeed draw epistemological objections from some philosophers and mathematicians, e.g. Tymoczko, who argue that they are not "real" proofs but rather preliminary calculations. As for simulations, they are not so much analogous to experiments as to thought experiments. The same objections apply, the underlying worry being that computers eliminate the element of insight involved in human proofs and thought experiments.
    – Conifold
    Sep 5, 2017 at 19:54
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    I think this should be split up into two different posts: Phil. Science and phil. math a very different topics and the effects of computers on each would warrant completely different approaches. Sep 6, 2017 at 4:03
  • Primarily opinion based as it is based on scientific speculation. The examples given have nothing to do with philosophy. Sep 7, 2017 at 6:51
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    It might have some significance in philosophy of science but my first thought is that in the general discipline of philosophy it has none. Most philosophical problems require only a knowledge of the first four numbers. (Nihilism, Monism, Dualism, Trinitarianism). Even the problem of the continuum needs only two numbers connected by a line. It;s hard to image what a philosopher would do with a computer as any more than a word-processor and internet connection.
    – user20253
    Sep 8, 2017 at 17:30

1 Answer 1


Since the epistemological and methodological issues are similar with simulations, which are computer extended analogs of thought experiments, and computer assisted proofs I will only focus on the latter. Computer assisted proofs came into prominence with the Appel-Haken-Koch's (supposed) proof of the Four-Color Theorem. Tymoczko in his 1980 paper Computers, Proofs and Mathematicians wrote:"Computer-assisted proofs illustrate the need for a more realistic philosophy of mathematics that allows for fallibility and empirical elements". But this meant extending the scope of mathematics only while demarcating the "rigorous", computer-free one:

"The lemma claims that each configuration in a certain set is D-reducible (or C-reducible). The grounds for this lemma are that suitably programmed computers delivered certain output when given certain input. However, such grounds are inherently fallible. The output could have been misread, the computers might have malfunctioned, the computers might have been misprogrammed, the programs might not have captured the mathematical intention. If any of these possibilities obtained, then, for all we know, the Four-Color Theorem might be false... To be sure, the possibility for error is rather small. It does not preclude mathematicians from knowing the Four-Color Theorem any more than the possibility of error prevents scientists from knowing facts about the physical universe. But that small possibility of error does preclude mathematicians from knowing the Four-Color Theorem with absolute certainty. The proof is not rigorous."

Since then it was pointed out that "absolute certainty" was never absolute, and issues with social acceptance of proofs did not originate with computers. For instance, Kempe's 1879 proof of the four color theorem was accepted as such for 11 years, until Heawood found a flaw in it. A fierce debate about the role of rigor and experiments/simulations commenced when Jaffe and Quinn in 1993 suggested to implement Tymoczko's idea by demarcating "theoretical" (modeled on theoretical physics) and "rigorous" mathematics institutionally, in publications, etc. See discussion and references under What makes something mathematics? The institutionalization idea did not take (some saw the Jaffe-Quinn proposal as an attempt to devalue "theoretical mathematics"), but clear delineation of conjectural/heuristic material remains an informal norm. It is also expected that any substantive use of computer tools, even in calculations let alone proofs, is explicitly mentioned and described enough to be reproducible (which is similar to reporting experiments in science).

But hand-wringing over "absolute" rigor is not the only concern. Hamming, whose mathematics was computer science oriented, is well known for quipping "typing is no substitute for thinking" and "the purpose of computing is insight, not numbers". One can take it pragmatically, or one can take it more fundamentally. Even pragmatically, simulations of the hurricane movements, say, may give us predictions, but not why they are thus and so. When models disagree we are left in the dark, because the crucial element of insight is missing. And on the Kantian/intuitionistic conception of mathematics and mathematical validity the chasm between human and computer assisted proofs is more than a matter of degree. That a deduction constitutes a proof as long as its every step is according to the "rules" is a formalist idea. But not all scientists or mathematicians are formalists. To some there is that elusive aspect of intuitive insight without which a proof is at best a blind calculation, with a stigma of inferiority attached. "Understanding" just does not take place outside of an understander, and a proof is not a proof until it is understood. Similar sensibilities affect reception of quantum mechanics, complaints about "shut up and calculate" are not uncommon.

In Remarks on the Foundations of Mathematics, even before the advent of computers, Wittgenstein identified a general phenomenon that feeds such worries. To make a computer capable of assisting in a proof a lot of linguistic conversions are required. We typically assume "transparency of language" and see them as converting the "same" calculations into different formalisms, but supposed "conversions" may alter the conception of what is done. The "dangerous, deceptive thing" about it is that "it makes the determination of a concept - concept formation - look like a fact of nature":

"A mathematical proof must be perspicuous... The proof must be a configuration whose exact reproduction can be certain... What convinces us -- that is the proof: a configuration that does not convince us is not the proof, even when it can be shewn to exemplify the proved proposition. That means: it must not be necessary to make a physical investigation of the proof configuration in order to shew us what has been proved.

[...] I want to say: one need not acknowledge the Russellian technique of calculation at all -- and can prove by means of a different technique of calculation that there must be a Russellian proof of the proposition. But in that case, of course, the proposition is no longer based upon the Russellian proof.

[...] It is not logic - I should like to say--that compels me to accept a proposition... when there are a million variables in the first two pairs of brackets and two million in the third. I want to say: logic would not compel me to accept any proposition at all in this case. Something else compels me to accept such a proposition as in accord with logic... I want to say: with the logic of Principia Mathematica it would be possible to justify an arithmetic in which 1000 + 1 = 1000; and all that would be necessary for this purpose would be to doubt the sensible correctness of calculations. But if we do not doubt it, then it is not our conviction of the truth of logic that is responsible."

In other words, Wittgenstein rejects the formalism/implementation distinction (which implicitly appeals to language transcendent platonist entities "expressed" in different implementations), and points out that "conversion" creates a new formalism. Since in the case of computers one of them is "perspicuous" and the other not we have a qualitative difference between the two, not merely a pragmatic one. Of course, he would not object to the shift of accepting new proofs ("philosophy leaves everything as it is"), but rather to the philosophical self-deception that the leap taken here is nothing but a cosmetic makeover.

Such a surreptitious shift from constructive to platonist proofs is what motivated the intuitionist pushback, but as Wittgenstein points out the self-deception involved is far more pervasive. The point affects even formalists, as such shifts may involve major changes in the rules of the "formal games" of mathematics. On the other hand, it would be silly even for intuitionists not to make use of computer assisted proofs pragmatically. After all, when done by competent and trustworthy professionals the degree of certainty they provide is far greater than from numerical checks of conjectures, which have a venerable history involving Euler and Gauss.

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