Sunday, 1 September 2013

Inconsistency in Mathematics

What are sometimes called "Epistemic Theories of Truth" have the generic form,
$S$ is true if and only if $S$ has epistemic property $P$.
For example:
  • $S$ is true if and only if $S$ could be verified, in principle.
  • $S$ is true if and only if $S$ belongs to the (unique) maximally coherent set of propositions/statements/beliefs.
  • $S$ is true if and only if $S$ would be accepted in the ideal limit of rational inquiry
Each is designed to ward off scepticism and fallibilism. The identification of truth with some epistemic feature of representations is an attempt to try to close the mind-world gap, between:
how the world is represented to be in cognition 
how it is.
For example, Kant's argument for Transcendental Idealism is based on trying to close this gap by identifying space, time and physical objects with representations. He says that "space is the form of external intuition", but this is a very strong idealist assumption, and inconsistent with modern physics and cognitive psychology. The form of external intuition may be Euclidean 3-space (an attempt to develop this into a detailed theory was worked out by David Marr), while space (more exactly, spacetime) may well be an 11-dimensional manifold, or some kind of orbifold, or a discrete causal set and so on. Modern physics tells us that the cognitive representation of space as 3-dimensional and Euclidean is an illusion. Therefore, space is not the form of external intuition. Or, to put it as an inconsistent triad,
(Cog science) The form of external intuition is 3-D & Euclidean.
(Physics) Space is not 3-D & Euclidean.
(Kant) Space is the form of external intuition.
The gap between how we represent (perceive, etc.) reality and reality was noticed by ancient philosophers and emphasized, in different ways, by Descartes, Hume and Popper.  The world may be very different from what it appears to be like (or even what it is accepted to be like). This gap or mismatch between representation and reality is the basic principle underlying all Realist views in epistemology, metaphysics, etc. In contrast, Idealism attempts to close this gap by identifying reality with mental representations: e.g., an Idealist (e.g., Brouwer, Heyting in mathematics) might insist that numbers are "mental constructions", or identify truth with "what can be verified", etc.

Connected to all this, there's a very interesting point made by Tim Chow on the FOM discussion list the other day, in relation to a question about the existence and prevalence of inconsistencies in the body of "accepted" mathematics:
For a start, it's not clear what you mean by "inconsistent systems actually yielding false theorems." For example, how is this different from just making a mathematical error? You make an error, producing a provably false statement. In effect, at that point you are starting to "work in an inconsistent system," because you are taking your false statement for granted. Then before discovering the mistake, you reach something that you decide is worth calling a theorem. Behold, an example of an inconsistent system actually yielding a false theorem! This is not just a hypothetical scenario. The published mathematical literature contains many provably false theorems, some of which have not been noticed because nobody has bothered to read the papers in question. Since it is standard mathematical practice to take published results as axioms, the mathematical literature as a whole is one giant inconsistent system which has yielded false theorems, and if you have a vivid imagination then you can easily picture some false theorem that *could* make a "bridge fall down," especially since your scare quotes imply that no actual bridges need to be harmed in the making of this imaginary world.
Abstracting a bit, Chow's premises are:
  • there are statements that are accepted as having been proved, 
  • but which have not been proved, because the alleged proofs weren't, in fact, proofs
The conclusion being that the body of published mathematical literature is, strictly speaking, inconsistent.

This point causes difficulties if one attempts to define "proof" in terms of "being accepted as a proof". Some text may be accepted as a proof, despite not being a proof; and, conversely, something might be a proof, while not accepted as one. Fallibilism runs deep, occurring even in mathematics, an area where epistemic standards are pretty high.

(I don't want to give the impression of disputing these standards! I assume that more obvious inconsistencies and errors are, typically, spotted and knocked out quickly. I know from experience that I notice mistakes in stuff I scribbled down yesterday, trying to figure something out. In working with other logicians, the same occurs all the time, in reading papers, attending talks, etc. Mathematical practice is deeply Popperian. Proofs are discovered by guesswork.)

Might the presence of certain, hard-to-isolate, inconsistencies in accepted mathematical literature lead to serious problems, bridges falling down, and whatnot? Chow adds this:
More to the point, if your response is that you're restricting your attention to explicitly articulated formal systems that have been widely adopted for use in science and engineering, then the problem is that scientists and engineers never work explicitly in any formal system. F.o.m. simply does not play that role in the real world. If a bridge actually falls down for some theoretical reason, it's going to be because the mathematical *model* fails to take something important into account, e.g.,
or for a less hackneyed example:
There was a somewhat extended discussion of this on FOM, involving the presence of software errors and database inconsistencies. I'm not sufficiently au fait with this to know much about the "real-world consequences". I do recall that, several years ago, NASA made a computational error (concerning S.I. units) with the Mars Climate Orbiter, causing it to crash.


  1. Read Wittgenstein (Frege, Ryle) on 'the mind' and the supposed mental character of representation.

  2. Thanks, Anon, but not sure I get this cryptic remark!
    I guess the suggestion is that Wittgenstein and Ryle argued that "minds" don't exist and there are no mental representations?

    But aren't belief states mental representations? Doesn't the visual system generate cognitive representations?