Bridgeman writes in A Physicist's Second Reaction to Mengenlehre:"The feeling that actually existing things are not self-contradictory is so elemental as to almost constitute a definition of what we mean by self consistent. Now when we are concerned with things we are evidently concerned with some form of experience, so that we may make an even broader statement and say that experience is not self-contradictory". This appears convincing at first, but the more you think about it the less sense it makes.

Historically, many experimentally successful physical theories were inconsistent, classical electrodynamics had infinite self-interacting energies, Planck's radiation and Einstein's photoeffect theories treated light as both waves and quanta, Bohr's atom was a chimera too, quantum mechanics must have "quantum objects" interacting with distinct "classical apparatus", which nonetheless consists of those same quantum objects, and incompatibility of the standard model with general relativity famously motivated the string theory. There is no detectable 'growth' of consistency either, if anything the conciliation difficulties increase.

First, it means that the logic of scientific theories is not classical, if it were the law of explosion (contradiction implies anything) would make them useless. Scientists obviously have ways of compartmentalizing contradictions. But even trying to justify a belief in some self-consistent "theory of everything" seems problematic to me. The reality is supposed to follow some "intelligent design", Platonic or theistic, that we are discerning from experience? This is worse than what it is supposed to justify. And even if there was such a consistent "intelligent design" science is a compromise between that and human limitations. We deal only with our best approximation, and that may well be a patchwork of contradictory but practically successful mini-theories rather than a "theory of everything".

Was the logic of scientific theories studied in how it deals with internal contradictions, are there formal models of it? Are there compelling empirical or philosophical reasons for scientific theories to be (eventually) self-consistent?

  • 1
    Re “There is no detectable 'growth' of consistency either”, on the contrary, among your listed examples Einstein's work on the photoelectric effect removed the classical infinity of black body radiation. And so Einstein got his Nobel prize, for that work (not for relativity). Also among your examples, some suitable generalization of string theory may (some expect it will) remove the incompatibility between relativity and quantum mechanics. The world progresses. :) May 23, 2015 at 0:11
  • @Cheersandhth.-Alf: The world does indeed progress but undoubtedly new differences will open up or be shown. May 23, 2015 at 0:39
  • @Cheers and hth. - Alf Half full vs half empty :-). The way I see it quantum mechanics resolved Planck's, Einstein's and Bohr's inconsistencies, but introduced far more intractable ones instead, and what string theory is attempting to do is beyond even that. Standard model is more than inconsistent, there is not even a formulation of it beyond the perturbation theory. The complexity will only grow, and so will the intractability. The progress is made in empirical coverage, not in consistency.
    – Conifold
    May 23, 2015 at 0:40

3 Answers 3


I'm unaware of any cases where formal mathematical inconsistency is not viewed as being due to, or indicative of, the limited domain of applicability of the theory.

Self energy of the electron? Well, Maxwell's equations were formulated in the 1860's; the manifest detection of electrons as discrete charge carrying particles didn't occur until 1897. Even after their discovery there was the possibly that they had a non-zero size, and thus everything in classical EM would be finite (technically this possibility of a non-zero size to electrons still exists, but would need to be handled with quantum electrodynamics). The mathematics of literal point charges can be used as an approximation in those cases where the size of the charge distributions are much smaller than any other length scale in the problem.

The Bohr model was quickly found to work for hydrogen (and if I recall correctly He+ ions and partially for alkali metals) but not generally across the elements -- it's applicability (at least now) is known to only be to cases where there is a single electron in an orbital (using somewhat more modern language).

The ultraviolet catastrophe indicates that the Rayleigh-Jeans law is (at best) an asymptotic description relevant for long wavelengths, and it is still applicable and useful in that domain.


To some extent I'm disagreeing with the characterization of the issues as "contradictions" -- science isn't just logic, so these issues aren't (necessarily) identical to logical contradictions. In particular they don't follow the law of explosion.

Another way to say it is: these theories are consistent if their (some times post-hoc) bounds of applicability are taken account of.

  • I more or less agree with your remarks, this is what I meant by compartmentalization and patchwork of mini-theories, domains of applicability are a nice way to explain how contradictory theories avoid explosion. But the question is whether there are reasons to expect any consistent synthesis of them, or to insist on one, beyond the practical preference because consistent theories are easier to use. If domains of applicability solve the logical problem then consistent uber-theorie, if one even exists, may not be even practically preferable.
    – Conifold
    May 27, 2015 at 2:16

We are drawn to classical logic, but we seldom use it in real life. When we approach a moral decision, or devise a system of law, we do not avoid conflicts, we presuppose methods for working them out in the long run. Common Law is a mass of contradictions to be negotiated according to two standards that are themselves contradictory, a statute prevails to the degree it is established and current, so the older it is, the better, and the more recent it is, the better.

There is no reason to think science should work differently, especially given that mathematics itself does so only by pretending it means something much clearer than it can. We can wish away Russel's paradox, but only if we become quite obsessive about avoiding it.

I am sorry to harp on the same gap over and over again, but I think this is another case where we wish to conflate all of science with the Kuhnian phase of normal science. The goal of normal science is to find ways to apply the current theories to observations in a way that makes the results predictable. So the focus is on consistency, but only in a secondary sense. As optics shows us, predictable inconsistency is still a solution -- light is both a particle and a wave and that is that, you just have to know which matters most in different situations.

But there is not a strong need for avoiding or resolving contradiction outside of normal science: which means between disciplines or between strong paradigms within a discipline, which comes down to all the time, to the extent normal science is never quite finished before another paradigm begins to affect at leat some leaders' thinking.

A lot of theories of science from the late 20th century, arising in contradiction to Popper, emphasize how theories can evade stringent logic when they have appropriate sociology or psychology behind them. Kuhn emphasizes this in terms of revolutionary periods, but people springing off of him like Lakatos and Toulmin look at the dynamics between theories more as convergence and conflict on an ongoing basis. And the token complete anarchist, Feyerabend drives us to consider a sort of meta-methodical way of looking at theories that does not attempt to force them to agree with one another.

The problem with thinking like this is that it offends scientists, who would really like to see their disciplines spend long periods of time in veins that aggree with Popper, and lets them compute precisely how much to question each result. To me this is an assertive idealization made in the name of realism, and it is very hard to shake.

  • I tend to agree with this; but having not spent any time with scientists is it fair to characterise their whole thinking with what ought to be only a part of it? May 23, 2015 at 23:22
  • 'the goal of normal science is to apply...observations to theories'; I suppose one could call this phenomenological aporias; in high-energy physics for example one has the phenomenologists that try to extract usable predictions to theories - is this along the lines you mean? May 23, 2015 at 23:25
  • I mean more directly Kuhnian 'puzzle-solving', to clarify your theory and make your predictions consistent with your observations. If it is really expensive to make any observations, you may have to work 'backward', but you are still trying to get the two to match when the data comes in.
    – user9166
    May 24, 2015 at 0:34
  • I am confused as to what the first comment means. I have spent considerable time with both 'hard' (biologists and physicists) and 'soft' (psychologists and cultural anthropologists) scientists. And I think the public's imagination and their own ego favors the notion that they are always doing 'normal science', when in fact, on an hour-by-hour basis they just aren't.
    – user9166
    May 24, 2015 at 0:45
  • I was referring to your last paragraph; now that I look at it I see that I wasn't particularly clear; I just meant is it fair to characterise how someone thinks (say a logician like Bertrand Russell) when they are exercising their professional expertise (in this case logic) with how they they when they are not (Russell wrote on other topics and we can see quite visibly that he isn't using 'formal logic' to think but thinking); if scientists are not doing normal science on a day-to-day basis then what are they doing? May 24, 2015 at 12:26

First, I assume that by a scientific theory you mean in fact a theory of physics; and we mean by this an actual one - one in which calculations can be made and predictions predicted.

Then until one obtains a final theory what we must have a patchwork of theories; as indeed we do today with the largest schism between QM and GR.

As the universe is a whole; one imagines a hidden order that is intelligible; and our present considerations shows that this is in part correct.

The question is whether a final theory which reveals the entire intelligible order is possible; or must it always remain an ideal; and in part this rests on what constitutes a physical theory; if one expression of this is correct - that it must be written in the language of mathematics; then a different line of attack opens which is the logical rigour of the mathematics; after all the calculus itself has founded itself on differing foundations - analytic, logical and synthetic.

  • I suppose hidden order is intelligible by definition, or we would not call it order, but just because we want it or imagine it does not mean it exists. Even as an ideal limit. I do not see what shows that this is correct even in part, in fact I do not see how anything can show that, considering that we have no way of getting out of ourselves and comparing what we have to an ideal, or verifying that we approach it in any sense. So no hidden order, if there is one we may never approach it, and even if we did we'd never know. I dislike the nihilism, but I do not see good arguments to refute it.
    – Conifold
    May 23, 2015 at 0:52
  • I don't regard it as a 'nihilistic'; something's are 'shown'; plus there is the notion of coherentism in truth justification. May 23, 2015 at 0:56

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