Jules Henri Poincaré (1854-1912) was one of the greatest mathematicians of his era and is sometimes said to deserve co-credit with Einstein for the discovery of Relativity. His interests also ranged into philosophical issues, in the philosophy of science and mathematics and even in the relation of science to morality. He is especially well known for his view that knowledge is based on conventions adopted by scientists and mathematicians, not on objectively determined features of reality. That kind of thing gets mentioned just in passing by people like Susan Haack (in Evidence and Inquiry, p. 99).
Poincaré's conventionalism is a mistake, but here I am more interested in two other citations and the kind of philosophical mistakes that they demonstrate. The first is found in a discussion of Kant in the popular philosophical novel Zen and the Art of Motorcycle Maintenance, by Robert M. Pirsig [Bantam, 1974]. Although this may not seem like a serious venue for discussion of Kant, it does reflect a misunderstanding that is all too common in more professional philosophy, as may be seen in The Ontology and Cosmology of Non-Euclidean Geometry.
To solve the problem of what is mathematical truth, Poincaré said, we should first ask ourselves what is the nature of geometric axioms. Are they synthetic a priori, as Kant said? That is, do they exist as a fixed part of man's consciousness, independently of experience and uncreated by experience? Poincaré thought not. They would then impose themselves upon us with such force that we couldn't conceive the contrary proposition, or build upon it a theoretic edifice. There would be no non-Euclidian [sic] geometry.
Zen and the Art of Motorcycle Maintenance, p. 236
The mistake found here concerns the nature of synthetic a priori propositions, with the conclusion that "we couldn't conceive the contrary proposition, or build upon it a theoretic edifice." What is striking about this conclusion is that it is the precise contradiction of the definition of a synthetic proposition. In Kant's terms, because the meaning of the predicate of a synthetic proposition is not already contained in the meaning of the subject, both the affirmation and the denial of the predicate are equally possible in terms of logic alone. Or, as Hume puts it, "The contrary of every matter of fact is still possible; because it can never imply a contradiction, and is conceived by the mind with the same facility and distinctness, as if ever so conformable to reality" [David Hume, Enquiries Concerning the Human Understanding and Concerning the Principles of Morals, Oxford, 1902, p. 25]. Hume, like Kant, divides all propositions into two exhaustive categories, "relations of ideas," which cannot be denied without contradiction (i.e. analytic), and "matters of fact," which can (i.e. synthetic). What is then revealing is that Hume considers geometry to belong to the "relations of ideas" category: "Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence" [ibid. p. 25]. The existence of non-Euclidean geometry refutes Hume, not Kant, though the vulnerability of Hume's theory in this respect is as little noted as the supposed vulnerability of Kant's is as often, falsely, stated.
If the axioms of geometry are synthetic, as Kant maintained, then this theory implies a prediction, not a denial, of the possibility of non-Euclidean geometry. However, Kant did not make such a prediction. Instead, he said that the axioms of geometry are known to be true a priori, and this is what confuses the issue. How could a synthetic proposition be true a priori? Of course, that is precisely the question Kant asks himself. His answer about geometry is that the structure of space is not given to us conceptually. The axioms of geometry do not "impose themselves upon us with such force that we couldn't conceive the contrary proposition," but a certain structure of space imposes itself upon us in "pure intuition," i.e. the imaginative construction of space. In short, non-Euclidean geometry can be conceived but cannot be visualized. This is still actually true: models and projections can be constructed of non-Euclidean spaces, but they all involve some kinds of distortions that would not be present in true spaces of that kind. Again, this is more thoroughly explored in The Ontology and Cosmology of Non-Euclidean Geometry.
The decisive mistake in both Poincaré and Pirsig's use of him is a failure to understand the meaning of the term "synthetic." But this is all too common.
The second citation of Poincaré that is of interest occurs in Leonard Nelson's System of Ethics [Yale, 1956]. The discussion, drawing on Poincaré's essay La Morale et la Science, is about whether ethics can belong to rational or "scientific" knowledge. Poincaré produces a familiar argument:
For if the premises of a syllogism, he [Poincaré] argues, are in the indicative mood, the conclusion must also be in the indicative, and thus cannot have the form of an imperative; yet to be part of ethics, it would have to take this form. For ethics deals not with matters than can be expressed in the indicative but only with matters that can be formulated in the imperative -- not with what happens but with what ought to happen. But science, says Poincaré, is always a system of propositions in the indicative; consequently, there can be no scientific validation of ethics."
System of Ethics, p. 4.
This is, of course, a reproduction of Hume's famous observation from A Treatise of Human Nature [Oxford, 1888], which runs as follows:
I cannot forbear adding to these reasonings an observation which may, perhaps, be found of some importance. In every system of morality, which I have hitherto met with, I have always remark'd, that the author proceeds for some time in the ordinary way of reasoning, and establishes the being of a God, or makes observations concerning human affairs; when of a sudden I am surpriz'd to find, that instead of the usual copulations of propositions, is, and is not, I meet with no proposition that is not connected with an ought or an ought not. This change is imperceptible; but is, however, of the last consequence. For as this ought, or ought not, expresses some new relation or affirmation, 'tis necessary that it shou'd be observ'd and explain'd; and at the same time that a reason should be given, for what seems altogether inconceivable, how this new relation can be a deduction from others, which are entirely different from it. But as authors do not commonly use this precaution, I shall presume to recommend it to the readers; and am persuaded, that this small attention wou'd subvert all the vulgar systems of morality, and let us see, that the distinction of vice and virtue is not founded merely on the relations of objects, nor is perceiv'd by reason.
David Hume, A Treatise of Human Nature, pp. 469-470.
Poincaré draws the same subjectivist conclusion as Hume, that morality is based on feeling, not any objective cognitive ground.
This kind of inference is common enough. The Logical Positivists seemed to think that all they had to do was cite Hume's argument and then propositions of ethics could be safely forgotten by serious (i.e. "scientific") philosophy. Doubts about the conclusiveness of this inference now often accept its form to the extent that it is regarded as necessary to find a way to derive imperatives from indicatives if the propositions of ethics are to be given cognitive dignity. Nevertheless, Hume's argument establishes nothing that could not already have been comfortably admitted by Aristotle, and what are often taken as parts of the conclusion of Hume's argument are really only hidden premises of the typical interpretation of it.
Aristotle would have had no difficulty because he already thought that each of the "sciences," i.e. deductive systems of knowledge, had its own first principles, i.e. its own axioms which cannot be proven in the system and cannot be proven in other systems. Aristotle explained their status by claiming that they were self-evident [cf. The Foundations of Value, Part I, Logical Issues: Justification (quid facti), First Principles, and Socratic Method]. Now, we may say that it was a great achievement of Hume and Kant to dismiss such claims of self-evidence (although Hume didn't, as Kant did, for geometry or arithmetic); but the denial of their self-evidence is today often carelessly coupled with a hidden premise: that if they are not self-evident, then they are not otherwise known to be true. This may even be true, but it does not follow from Hume's argument. There may be other ways in which the "first principles of demonstration" may be true, and be known to be true, without being self-evident. That was the whole point of Kant's theory of synthetic a priori propositions, though he fails to state it in terms of the problem of first principles. Unless it is shown that self-evidence is the only way that first principles can be known to be true, or be true, the cognitive status of the imperative axioms of ethics is in no way threatened by Hume's, or Poincaré's, argument.
Indeed, the project of proving that only self-evidence could justify first principles is rarely even conceived, much less undertaken. Similarly, neither is it noted that another hidden premise commonly occurs in the evaluations of Hume's conclusion: that is it acceptable for first principles, even if not self-evident, to be indicatives, but not for them to be imperatives. This would seem to be an assumption that imperative axioms are problematic, while indicative axioms are not. Stated thus baldly, this is obviously false; but it seems to be part and parcel of the strategy that the cognitive status of ethics can be dismissed simply by showing that indicative and imperative deductive systems must be axiomatically independent. This is an especially paradoxical move, under the aegis of Hume, when one notes that Hume's separation of moral axioms from reason precisely paralleled his separation of causality from reason: "All reasonings concerning matter of fact seem to be founded on the relation of Cause and Effect" [Enquiry, p. 26] -- even while Hume's greatest fame lies in his demonstration that inferences from cause to effect are not self-evident and are not founded on any other rational ground that Hume can identify: "The question still recurs, on what process of argument this inference is founded?" [p. 37]. Thus, if the propositions of ethics have no cognitive foundation, then no "reasonings concerning matter of fact" have a cognitive foundation. That is no less than what Hume himself thought, since he was a skeptic.
Although many moderns would actually agree with Hume's scepticism, it is still a non sequitur to hold ethical propositions at some disadvantage vis à vis factual ones. As Hume himself did not, it would be incumbent on those using him to explain the difference -- but it was typical of schools like Logical Positivism to lamely offer an inductivist justification of scientific knowledge, even when aware that Hume himself had also destroyed the cognitive force of induction. At some point, simple confusion would seem to give way to dishonesty, if we are to judge that many of the Positivists must have been educated enough to know the context of the arguments in Hume that they selectively used or ignored.
We need not so belabor Poincaré, who had probably not gone into all the details of the historic development from Aristotle to Hume and to Kant. When professional philosophers misunderstand and misuse Hume, mathematicians branching out into philosophical issues are at a grave disadvantage. The veneration with which many 20th century philosophers have held mathematics, however, then can serve to perpetuate and reinforce the mistake, as Poincaré's reiteration of the point may become authoritative in itself.
Nelson, of course, in the passage cited, does not make Poincaré's mistake, but exposes it. "Poincaré did not even investigate, let alone resolve, the question as to whether such a science [of ethics], consisting only of imperatives, is possible" [p. 4-5]. But then Nelson is not well enough known in 20th century philosophy for this exposure to have had its deserved effect on all the many cases when Poincaré's kind of mistake is endlessly repeated, and hidden premises, not Hume's classic argument, beg the question of the cognitive nature of ethical propositions.