Underdetermination, Logic, Mathematics and Science

One of the clearest examples of what Putnam tried to show is the case of Quantum Logic (Putnam 174, 248). Quine in “Two Dogmas” also alludes to Birkhoff and von Neumann, and their development of Quantum Logic as a possible instance of revision of logic using empirical basis (Quine 1953, 43). These are cases in which, supposedly, formal classic logic does not apply, and one of these seem to be quantum mechanics. Curd and Cover give this example that concerns the principle of the excluded middle. For example, both of these formulas are tautologically equivalent, one can be derived from the other:

1. p ∧ (q₁ ∨ q₂)

2. (p ∧ q₁) ∨ (p ∧ q₂)

This is true in classic logic, but not so in Quantum Logic. For example, let us take into account the two-slit experiment. Let p be “The elctron is in region R of the screen”, q₁ is the proposition “The electron went through slit 1”, and q₂ is the proposition “The electron went through slit 2”. If the electron goes through slit 1 or 2, we do not see any wave interference pattern formed on the screen. However, if they go through both slits simultaneously, then we can see the pattern. Therefore, the inference of (1) to (2) would be invalid in Quantum Logic (Curd and Cover 380).

However, one philosopher who refuted this Quine Putnam Thesis was Quine himself in his later book Philosophy of Logic. He argues that alternative logics (such as Quantum Logic) do not really revise classic logic. In Quantum Logic, the connectives are not defined in the same way in terms of truth values as in two-valued logic. Hence, connectives such as negation, the conjunction, or implication in classic logic do not mean the same thing as in Quantum Logic. This can be an alternative logic alright, but it does not seem to revise classic logic, because the meaning of logical propositions is very different. Quantum Logic would be itself another logical system besides classic logic, not its replacement or revision (Quine 1970, 83-84). This position is very distant from the Quine of “Two Dogmas”, in which he contemplates the possibility of revision of logic and mathematics

All of this is related to the issue of whether there is or there is no a priori knowledge. Though Putnam is open to the idea of revisability in logic and mathematics, there can be a priori statements that seem not to be revisable. For instance, propositions like “2+2=4” leave no room for doubt at all, but propositions that seem “quasi-empirical”, may be revised, e.g. “Peano arithmetic is 10⁽²⁰⁾ consistent.” These “quasi-empirical” statements seem to be revisable if a contradiction is found (Putnam 124-126; Fred 132-134). Putnam seems to equate revisability with empirical experience, and, in his mind, empirical experience somehow implies revision of a priori knowledge (Fred 135). However, Bob Hale has pointed out, a prioricity is not incompatible with revisability (Hale 143).

Putnam, in implicating revisability with empiricism, is not clear on what the nature of his revisability in mathematics and logic is. For example, Platonists and realists in general do not equate a prioricity with infallibility of knowledge. There are mathematical truths that we know infallibly, such as the theorem that the square root of two is an irrational number. But there are also some mathematical conjectures that still remain undecidable, reason by which philosophers like Philip Kitcher has rejected mathematical realism in general (Kitcher 36-48). James R. Brown does not find fallibilism incompatible with Platonism, nor with a priori knowledge. Brown says that the fact that we do not know if certain mathematical conjectures are true, does not mean that ontologically speaking they are not true nor false. If we concede that non-discovery implies non-existence, this would be equivalent to say that in 1700 no one knew that there was a planet called Neptune, and therefore Neptune itself did not exist at the time. Though there is a priori knowledge, that does not mean that we have infallible a priori knowledge of everything. Fallibility in mathematics can stem from three main sources:

1. We could formulate false mathematical or logical conjectures, but not know they are actually false;

2. We could make mistakes that stem from incorrect application of accepted principles (wrong calculations;

3. We could use wrong and naïve concepts (Brown 1999, 18-23).

I would add a fourth aspect of fallibility in the field of mathematics, and that is that we do not know the entire abstract reality that we have not yet discovered. Notice that these sources of fallibilism have nothing to do with experience, but have to do more with the way we deal with the theoretical aspect of mathematical truths as a whole. The same goes with logic.

Some people might state that there were in fact instances that science actually revised mathematics, and that the General Theory of Relativity is one example of these. According to them, Einstein “proved” that space-time is non-Euclidean, while scientists before him thought that space was exclusively Euclidean. For them, General Relativity “proved” that Euclidean Geometry is false. First of all, Non-Euclidean Geometry was first formulated within mathematics, not within science. Lobachevsky, Riemann, and Bolyai developed different geometrical models that denied the “axiom” of the parallels but remained logically consistent, hence conceiving many possible geometrical spaces. Under these new geometries, Euclidean geometry was not refuted, but became one of an infinity of conceivable spaces in mathematics (Brown 1990, 101-102). and indeed this represented a revision in mathematics, but Euclidean Geometry itself was not revised. It only revised the conception of Geometry that admitted Euclidean space as the only one which is valid and possible. Also, notice that while these Non-Euclidean Geometries were developed, empirical criteria played no role in them. Many scientists and mathematicians during the XIX century philosophers rejected all Non-Euclidean Geometry as being non-empirical because space, for them, was Euclidean. For them, simply Non-Euclidean Geometry was entirely false, and had no potential in being applied to science.

Poincaré put an end to this conception of Non-Euclidean Geometry. He was the first one to conceive the logical possibility that Non-Euclidean Geometry could be adopted by scientists with the purpose of simplifying t theory. Poincaré dismissed this possibility because he regarded it as highly improbable. For him, Euclidean Geometry is in itself much simpler than Non-Euclidean Geometry (Poincaré 72-88). It was not until Einstein, inspired by Poincaré, who adopted Non-Euclidean Geometry as a solution for a problem concerning the Theory of Relativity. The dilemma that Einstein confronted was this:

1. If Euclidean Geometry is adopted to solve the problem of Special Relativity, then it complicates the scientific theory at hand. It would be very difficult to relate gravity, light speed, and the consequences of Lorentz Transformations to space-time.

2. If Non-Euclidean Geometry is adopted, though it is more complicated than Euclidean Geometry, it simplifies the scientific theory in great measure, and can give an effective account on how gravity works and its relation to space-time (Einstein 97-126).

As we can see here, Einstein, nor any other scientist, “discovered empirically” that space time is actually Non-Euclidean, nor did it mean a revision in mathematics in any way. The problem here had to do with mathematical models for the theory. Under one model (Euclidean Geometry), the theory would not work as well as in the case of the other one. In this sense, Non-Euclidean Geometry represented a more complicated mathematical model, but it was used to formulate a simpler theory that could account for empirical phenomena (Brown 1990, 103; Brown 1999, 86; Rosado 268). This did not diminish at all the validity of Euclidean Geometry, it just recognized the validity of Non-Euclidean spaces, and its genuine application to science. In no instance we see that the validity of these models depend on experience, and none of them have been refuted by it.

To sum it up, the First Quine-Putnam Thesis is false. Experience cannot refute any mathematical and logical truth. In fact, all of these logico-mathematical truths to be proven true, they do not need any authority of experience. To prove the irrationality of the square root of two, or to prove the Pythagorean Theorem, we only need to use mathematical notions (which themselves are not found in experience) like numbers, points, lines, figures, and so on, and axioms and theorems, such as the properties of the division and multiplication, or the addition of angles of a triangle in Euclidean Space, among others. Sensible experience, nor any scientific notion (such as mass, electrons, light, and so forth) play any role in these demonstrations. Even if we want to accept deviant logics based on scientific data, such as Quantum Logic, they would not imply a revision of logic. So, mathematics is a field completely independent from science, as logic is its own field also. Mathematics and logic together constitute a priori knowledge, and they are never refuted a posteriori.

I wish to add also, for those who argue the adoption of Quantum Logic on empirical grounds, that in reality Quantum Logic has not served to explain quantum behavior. All it does, apparently, is to shift the mystery from Quantum Physics to Logic, but unlike Non-Euclidean Geometry and General Relativity, such Quantum Logic as a logical model has not provided any more explanatory value to science (Curd and Cover 380). It is on the grounds of logic that many scientists and philosophers have rejected the Copenhagen interpretation of Quantum Physics, and consider the possibility of another quantum theory we have not yet formulated.