Non-Euclidean Geometry as a Counterexample to Millian Empiricism

Originally written for PHIL 1300 Philosophy of Mathematics with Prof. Joshua Schechter.

As a through-and-through empiricist, J. S. Mill believes that the foundations of mathematics are conclusions we arrive at by inductive reasoning. He admits that there is much deductive reasoning in mathematics, but the original source of all of these deductive mathematical conclusions is some list of truths we can describe as axioms—statements that do not require proof. Mill then claims that it is these axiomatic truths that are actually supported inductively. However, it certainly seems that these axioms are true a priori, or at the very least that any support they do have is not inductive. How can Mill account for this discrepancy?

First, it is clear that our axioms—so long as we restrict ourselves to the axioms of a simple field like geometry—do indeed receive what may appear to be a form of inductive support. The proposition that, for example, no pair of lines may enclose a space, is given inductive support by our continual observation that our real life “examples” of pairs of lines (that is, physical approximations of lines) can only enclose a space when they deviate extremely strongly from our conception of what it is to be a line. Furthermore, we encounter physical geometric objects so frequently, and they conform to these axioms so unfailingly, that we cannot help but be overwhelmed with inductive evidence, producing a feeling of certitude.

What’s more, according to Mill, is that this sort of visual observational evidence is not the only sort of induction we are performing when it comes to these axioms. Mill suggests that the unique property of the domain of mathematics compared to other sciences may be that it is so well represented by our mind’s eye (as opposed to it being a deductive rather than inductive activity.) Thus, since by simply picturing a pair of lines and concluding from an analysis of this mental image that they could never enclose a space, we are gaining inductive support for the claim. In addition, this procedure of inductively reasoning from mental pictures is not a purely mental process, since the knowledge that mathematics has this special property in which mental pictures constitute good evidence is an experiential and observational fact about mathematical objects. That is, we could attempt to perform this same process of reasoning inductively on mental pictures in another domain, e.g. in an attempt to answer the physics question of which objects float and which sink in water. This would be a futile exercise, however, since our mental pictures do not contain enough information in the domain of physics, and this very fact of futility is something we know from experience (and likewise for the fact that mental pictures would work for a geometrical question.)

To Mill, this onslaught of inductive evidence not only continuously forces itself upon us, but is also inseparable from the mathematical concepts themselves. Mill tells us that “in the process of acquiring the idea we have learned the fact” and, quoting Alexander Bain, claims “when we have mastered the notion of straightness, we have also mastered that aspect of it expressed by the affirmation that two straight lines can not inclose [sic] a space.” Thus,

Mill extensively defends against claims that our axioms are necessarily true, a claim that Mill contends arises out of the inconceivability of denying them. Since, according to Mill, our notions of the concepts are so deeply tied in with our inductive conclusions about them, the inconceivability of the axioms’ denial is a property of our experiential relationship with the axioms and the world, rather than any logical property regarding the necessity of the axioms. In other words, “, in the case of a geometrical axiom, such, for example, as that two straight lines can not inclose a space—a truth which is testified to us by our very earliest impressions of the external world—how is it possible… that the reverse of the proposition could be otherwise than inconceivable to us?”

However, this counterargument Mill preemptively defends against contending that geometric axioms are necessarily true would seem silly in the modern day, precisely because it is no longer inconceivable to imagine their negation. A decade before Mill wrote A System of Logic, Nikolai Lobachevsky and János Bolyai had begun developing a branch of mathematics which would over the course of the next century become known as hyperbolic geometry, the system that results when one denies the Euclid’s parallel postulate (to be precise, the Playfair postulate that stipulates that for a line and a point off the line exactly one parallel line can be drawn through the point is denied, and substituted for a postulate claiming that in fact multiple lines can be drawn.) This consideration leads to a strong objection to Mill’s theory: its picture regarding situations like these non-Euclidean geometries, in which an axiom has been negated, is poor.

Clearly, the new axiom of hyperbolic geometry is not in any way inductively supported. However, the geometric world that it generates is indeed conceivable (with the aid of various models.) It is no longer clear exactly what Mill would say about the process of doing geometry, if we take the hyperbolic axioms. Does the faculty of mental visualization here give inductive evidence about hyperbolic figures? If not, is there some experiential way to determine this, as with physics? What exactly are hyperbolic figures? They are not idealizations of real world approximations as Mill describes Euclidean figures, but neither can they be mental objects since they face the same problems Mill ascribes to Euclidean figures in this regard (we can no more easily imagine a hyperbolic line without breadth than a Euclidean line.)

In particular, an objection to Mill may go as follows: doing Euclidean geometry (that is, describing axioms and definitions, proving theorems, etc.) and doing hyperbolic geometry look to be the same sort of activity, but on Mill’s account the former is an inductive science while the latter is a pure fiction. Mill may invoke a similar kind of argument as he does regarding the study of an “imaginary animal” in that “the conclusions which we might thus draw from purely arbitrary hypotheses, might form a highly useful intellectual exercise: but as they could only teach us what would be the properties of objects which do not really exist.” However, unlike in the case of an imaginary animal and a real animal, the objects of Euclidean geometry and the objects of hyperbolic geometry seem to be of the exact same nature: neither one describes an object that truly exists in the world, each is governed by the axioms we presuppose. When we discuss imaginary animals versus real animals, the semantics of our two activities are clearly wildly different (no one could possibly conflate writing novels about anatomically correct winged Pegasus horses with studying the physiology of real horses.) That is, we do not talk about hyperbolic and Euclidean geometries in a different way, we do not seem to describe a fictional world in one case and the real world in the other. We would need totally different semantics of these two identical seeming types of propositions in order to account for Mill’s claims.

An even more devastating point for Mill is that it is actually a matter of contention in cosmology whether the curvature of the Universe is exactly zero, or whether it is some very small positive or negative number. If the curvature of the Universe were nonzero, it would mean that the geometry of the Universe would not be perfectly Euclidean (i.e. even an impossibly perfect triangle would have an interior angle sum not equal to 180 degrees.) Does this mean that our barrage of induction is wrong? Even more worryingly, does this mean that if it were confirmed that the curvature of the Universe is some very small negative number, suddenly the semantic and epistemic status of all Euclidean geometry changes its nature drastically?

The consideration of hyperbolic geometry and other negations of axioms seems to me to be an insurmountable objection to Mill, who deeply relies on the inconceivability of such negations. If Mill commits himself to the idea that not only are mathematical objects by their nature solely idealizations of real life objects, but also that this is what we mean when we discuss them, then the fact that the degree to which these objects actually serve as limit objects of the real world is actually up for debate should serve to shatter all conversations around mathematics. If he decides to describe mathematical objects (either metaphysically or semantically) in any other way, e.g., that all statements about them are vacuous, he would be forced to accept that there is a parity with the objects of hyperbolic geometry. There is already a way to describe generalizations of real life geometrical figures, e.g., “as long as we draw a pair lines straight to a good enough approximation, they will never enclose a space.” It seems clear that the meaning of this sentence is distinct from what we mean when we say “No pair of lines can enclose a space.” In particular, in the second sentence there is an implied first clause: “(Given the axioms of Euclidean geometry,) no pair of lines can enclose a space.”

That is, if we object to Mill not that the axioms of Euclidean geometry are unimpeachable, necessary, a priori truths, but rather merely antecedents to the conditional in every mathematical statement, he has no recourse. Of course, the reason we came to choose these axioms and not those of hyperbolic geometry is because of a generalization of experience. However, this process no more gives the theorems of Euclidean geometry any semantic or epistemic upper hand over those of hyperbolic geometry than the historical research that goes into writing gives a historical fiction novel about 1800s America an upper hand over a science fiction novel about aliens. That is, the historical fiction novel is no more true, the characters and plots are just as fictitious, despite the fact that the axiomatic premises that govern the world of the novel were arrived at through an idealization of the real world.

Mill is therefore unequipped to handle this challenge. He can connect Euclidean geometry to experience, but the uncertain nature of real-world geometry (curvature of the Universe), the elimination of arguments around inconceivability, and our counter-theory in which that experience governs our choice of axioms but does not confirm them to be true put severe strain on Mill’s account.