Hypothesis

Plato seems to think that the mathematical sciences are hypothetical in their foundation, and that only in the science of dialectic, which he considers the highest science, does the mind rise from mere hypotheses to the ultimate principles of knowledge. "The students of geometry, arithmetic, and the kindred sciences," Socrates says in the Republic, "assume the odd and the even, and the figures and the three kinds of angle and the like in their several branches of science; these are their hypotheses, which they and everybody are supposed to know, and therefore they do not deign to give an account of them either to themselves or others."

There is, however, a higher sort of knowledge, "which reason herself attains by the power of dialectic, using the hypotheses not as first principles, but only as hypotheses, that is to say, as steps and points of departure into a world which is above hypotheses, in order that she may soar beyond them to first principles." The method of hypothesis appears earlier in the Meno, where Socrates proposes to investigate whether virtue can be taught by first hypothesizing that virtue is knowledge, and in the Phaedo, where one posits the strongest available hypothesis and examines whether its consequences cohere. In each case, the hypothesis is treated as a provisional assumption, to be pressed further or abandoned if its consequences fail.

What is most characteristic of Plato's treatment is the conviction that hypotheses are never endpoints. A mind content to rest on unexamined assumptions remains in the lower segments of the Divided Line, and the philosopher's task is to press through every hypothesis until reaching what needs no further support. The question of whether such unhypothetical first principles can be attained, and whether the mathematical sciences rest on foundations that are genuinely provisional, is one that Aristotle addresses by distinguishing between hypotheses and axioms in a way that differs from Plato's usage. This issue is treated more fully under the ideas of Principle and Metaphysics.

The issue between Plato and Aristotle may be, as some interpreters suggest, partly verbal, a difference in the use of such words as "science" and "dialectic." Whether it is verbal or real, it throws light on the difference between a hypothesis as a merely provisional assumption, susceptible to proof by higher principles, and a hypothesis as a probability taken for granted for the purposes of argument. Aristotle will attempt to resolve this by classifying the starting points of demonstrative science into axioms, definitions, and hypotheses proper.

Aristotle defines scientific knowledge in terms of three elements, one of which consists of the primary premises upon which demonstrations rest. The principles of a particular science may be axioms in the strict sense of being self-evident truths and hence absolutely indemonstrable; or they may be provisional assumptions which, though not proved in this science, can nevertheless be proved by a higher science. Hypotheses, in Aristotle's usage, are distinguished from axioms and from definitions: they posit the existence of the subjects a science studies, as when the geometer assumes that points and lines exist, and they are accepted without demonstration as conditions of inquiry.

This classification addresses what Aristotle regards as a difficulty in Plato's account. Plato wished dialectic to transcend all hypotheses and reach an unhypothetical first principle. Aristotle holds that each science has its own proper starting points, and it is not the business of geometry to justify the existence of spatial magnitudes. That belongs to a higher science, first philosophy. The hypotheses of a particular discipline are not defects in reasoning but necessary conditions for it. The distinction between scientific reasoning, which rests on true and primary premises, and dialectical reasoning, which rests on opinions generally accepted, is central to Aristotle's account. Dialectical reasoning "moves entirely within the sphere of opinion," and even the best opinions remain probabilities, propositions which are not self-evident and which cannot be proved. The question of the foundations of mathematics, treated more fully under the idea of Mathematics, is closely related to this analysis.

Reasoning which rests either on axioms or on demonstrable principles Aristotle calls scientific, but reasoning which rests only on hypotheses he regards as dialectical. This distinction has considerable consequences for the organization of knowledge. By making hypotheses relative to a particular discipline, Aristotle makes possible the independence of the special sciences. Each science is entitled to its own assumptions, as in "the application of geometrical demonstrations to theorems in mechanics or optics, or of arithmetical demonstrations to those of harmonics."

Aristotle's distinction between hypotheses, definitions, and axioms became the standard vocabulary for discussing scientific foundations. Aquinas will apply this framework to the question of whether the articles of faith can function as the hypotheses of sacred doctrine, while Bacon and the early moderns will insist that hypotheses must answer to experiment rather than to syllogistic authority.

Aquinas applies Aristotle's account of scientific principles to the question of whether sacred doctrine is a genuine science. Like every science, theology begins from first principles that it does not itself demonstrate. In geometry, those principles include hypotheses about the existence of points and lines. In theology, they are the articles of faith, revealed by God and accepted on divine authority. Sacred doctrine "does not argue to prove its principles, which are the articles of faith, but from them it proceeds to prove something else," just as the geometer proceeds from the existence of the triangle to the demonstration of its properties.

The principles of a particular science may be, as Aristotle observed, provisional assumptions which, though not proved in this science, can nevertheless be proved by a higher science. Aquinas holds that theology is a subalternate science in precisely this sense: it borrows its principles from a superior discipline, the knowledge God has of himself, which is communicated through revelation. The believer accepts the articles of faith the way the student of optics accepts the axioms of geometry, on the authority of the higher science. This allows Aquinas to maintain that theology is rational, its method demonstrative and its conclusions binding, without requiring that its foundations be established by unaided reason. The question of the relation between faith and reason, and of the foundations of theological knowledge, is treated more fully under the ideas of Theology and Knowledge.

Whether the articles of faith can function as genuine hypotheses in the Aristotelian sense has been a matter of considerable controversy. A hypothesis in Aristotle's framework is a provisional assumption that may in principle be demonstrated by a higher science; but the articles of faith, being matters of divine revelation, are not susceptible to proof by any human science. The question thus arises whether calling revealed truths "hypotheses" extends the Aristotelian framework beyond its original intention or whether it represents a legitimate application of that framework to a new domain.

Aquinas preserves Aristotle's account of scientific structure while extending it into territory that the original framework was not designed to address. Bacon and the early moderns will insist that hypotheses must answer to experiment rather than to any form of textual or theological authority, a demand that represents a considerable departure from both Aristotle's and Aquinas's understanding of the foundations of knowledge.

Bacon insists that hypotheses are not to be settled by authority or syllogistic reasoning but by methodical observation and experiment. A well-constructed experiment, especially what he calls an experimentum crucis, derives its demonstrative character from the hypothetical reasoning which formulates the problem to be solved. The value of such a crucial experiment appears, for instance, in Bacon's reasoning about the rise and fall of the tides: "If it be found that during the ebb the surface of the waters at sea is more curved and round, from the waters rising in the middle, and sinking at the sides or coast, and if, during a flood, it be more even and level, then assuredly, by this decisive instance, the raising of them by a magnetic force can be admitted; if otherwise, it must be entirely rejected."

The old logic, Bacon argues, flies from a few instances to the most general axioms and then deduces everything downward. This procedure generates hypotheses that flatter the mind's prejudices rather than tracking how nature operates. The four Idols catalog the systematic ways the understanding distorts its own hypotheses before experience has a chance to correct them. The remedy is a new method of induction: the investigator assembles tables of presence, absence, and degree, then eliminates candidate explanations one by one. Not all scientific work, as Bacon recognizes, is directed or controlled by hypotheses, "but in the absence of well-formulated hypotheses, the research can hardly be better than exploration." The question of scientific method, and of the relation between hypothesis and observation, is treated more broadly under the ideas of Science and Induction.

By making experiment the judge of hypotheses, Bacon shifts authority from texts and traditions to repeatable procedures. He does not think hypotheses are unnecessary; he thinks they are dangerous when left untested. The mind must propose explanations, but it must also discipline itself to let nature overrule its proposals.

Bacon's insistence that hypotheses must answer to experiment rather than to philosophical or theological authority had practical consequences that extended well beyond his own lifetime. Newton's later declaration that he does not "feign hypotheses" echoes Bacon's suspicion of speculative reasoning, though Newton's actual practice involves a more sustained engagement with mathematical theory than Bacon's method envisioned.

Descartes refers to certain matters assumed in his Dioptrics and Meteors, and expresses his concern lest the reader should take "offence because I call them hypotheses and do not appear to care about their proof." He explains that he has "not named them hypotheses with any other object than that it may be known that while I consider myself able to deduce them from the primary truths which I explained above, yet I particularly desired not to do so." The distinction between what is truly known and what is merely hypothesized is thus central to Descartes' philosophical method.

The method of doubt, which Descartes proposes in the Discourse on Method, may itself be understood as a hypothesis carried to its extreme: suppose that everything previously believed is false, and examine what, if anything, survives this supposition. The cogito, "I think, therefore I am," is what remains when every other assumption has been removed, and it is not itself a hypothesis but the foundation on which all subsequent knowledge is to be rebuilt. The rebuilding proceeds by a strict rule: only what is perceived "clearly and distinctly" may be accepted. Descartes derives the existence of God from the idea of perfection, and the divine benevolence guarantees that clear and distinct perceptions are reliable. The question of clear and distinct ideas as criteria of truth is discussed more fully under the idea of Truth.

In his physics, however, Descartes freely employs hypotheses that he admits may be merely probable. The vortex theory of planetary motion, for instance, is a hypothesis in the special sense that Newton will later criticize: a speculative construction designed to explain phenomena mechanically. The tension between Descartes' foundationalist aspirations in metaphysics and his use of provisional hypotheses in physics illustrates the difficulty of maintaining a sharp distinction between what is certain and what is hypothetical.

Newton will criticize the Cartesian vortices on the ground that it is unnecessary to appeal to unobservable entities in order to explain natural phenomena. The Cartesian hypothesis, like the substantial forms of Aristotle, is for Newton a hypothetical entity, not inferred from the phenomena. The question of the relation between hypotheses and the realities they purport to explain, whether hypothetical constructions are useful fictions or genuine representations of nature, is one that runs through the entire modern discussion of scientific method.

In the Mathematical Principles of Natural Philosophy, Newton says, "I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses; for whatever is not deduced from the phenomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy." The context of this passage, and of a similar statement at the end of the Opticks, seems to indicate a special meaning of "hypothesis."

Newton criticizes the vortices in the physics of Descartes on the ground that it is unnecessary to appeal to occult or unobservable entities in order to explain natural phenomena. The Cartesian vortices, like the substantial forms of Aristotle, are for Newton hypotheses in a very special sense: they are hypothetical entities, not inferred from the phenomena. Although treated as if they were realities underlying the phenomena, they are, as Gilbert says of the primum mobile, "a fiction, something not comprehensible by any reasoning and evidenced by no visible star, but purely a product of imagination and mathematical hypothesis." To make hypotheses, in the sense in which Newton excludes them from experimental philosophy, is to give reality to a fiction or construction of the mind.

Newton insists that the principles of physics should be deduced from the phenomena by induction, not proposed as conjectures. A principle deduced from phenomena carries the authority of the observations themselves; a hypothesis, however plausible, remains a conjecture. In practice, however, Newton's Opticks contains many "Queries" that are plainly speculative hypotheses about the nature of light, matter, and force. The tension between Newton's official methodology and his actual practice illustrates the difficulty of excluding hypotheses entirely from experimental science, a difficulty that Hume and Kant will address in different ways.

Hume will ask whether even Newton's inductive principles rest on an undemonstrable assumption about the uniformity of nature. Kant will argue that the mind contributes structural principles that make Newtonian science possible in the first place. The question of whether hypotheses are to be eliminated from science or disciplined within it remains one of the central issues in the philosophy of scientific method, discussed also under the ideas of Science and Physics.

Hume accepts the empiricist demand that hypotheses be grounded in experience, but he raises a difficulty that Newton himself did not address. All reasoning about matters of fact, Hume argues, rests on the relation of cause and effect, and all reasoning about cause and effect rests on experience. But experience only tells us what has happened, not what must happen. The inference from past to future depends on the assumption that nature operates uniformly, that the future will resemble the past. This assumption cannot be demonstrated by reason, since its denial involves no contradiction, and cannot be confirmed by experience, since any appeal to past experience already presupposes it. The foundation of all empirical hypothesis is itself a hypothesis without rational warrant.

Hume's point is not that hypotheses are useless or that science is fraudulent. It is that the practice of forming and testing hypotheses rests on a deeper expectation of regularity that cannot be justified by the very methods it underwrites. We believe the sun will rise tomorrow not because we have a proof but because custom and habit produce that expectation in us. The "problem of induction," as it came to be called, does not paralyze inquiry; scientists continue forming hypotheses and testing them. But they do so on a foundation that philosophy cannot secure. The question of necessary and contingent truths, and of the ground of causal reasoning, is treated more fully under the ideas of Necessity and Contingency and Cause.

This puts every empirical hypothesis in a peculiar position: it can be tested against experience, but the authority of experience itself rests on something that is not a finding of experience. Probability rather than certainty is the best that the testing of hypotheses can achieve, and this limitation is inherent in the logical structure of inductive reasoning rather than in the imperfection of any particular investigation.

Kant will respond to Hume by arguing that the uniformity of nature is not a psychological habit but a transcendental precondition of experience itself. Mill will attempt to formalize the logic of induction in a way that acknowledges Hume's challenge while preserving the capacity of hypotheses to yield probable knowledge. Both take Hume's analysis as their starting point.

Kant responds to Hume by distinguishing between what hypotheses can do in natural science and what they can do in philosophy. In natural science, hypotheses are legitimate instruments of investigation: one proposes a law, deduces observable consequences, and tests them. The uniformity of nature that Hume found problematic is, for Kant, a transcendental precondition of experience itself. The mind imposes causal regularity on appearances as a condition of having any coherent experience at all. Hume was right that experience alone cannot justify the principle of causality, but wrong to conclude that it rests on mere habit. It rests on the structure of the understanding. The question of the categories of the understanding, and of synthetic a priori knowledge, is treated more fully under the ideas of Knowledge and Mind.

In philosophy, however, hypotheses occupy a more restricted role. Kant argues in the Doctrine of Method that reason generates ideas, such as God, freedom, and the soul's immortality, that cannot be confirmed or refuted by any possible experience. These ideas do not constitute knowledge. They function as regulative principles, orienting inquiry by giving it direction and systematic unity. "The hypothetical use of reason, based upon ideas regarded as problematic concepts, is not properly constitutive," Kant writes; it does not establish the truth of the universal rule assumed as hypothesis. The idea of God as the ground of all order, for instance, does not prove that God exists but directs the investigator to seek the greatest possible unity in nature.

The distinction between constitutive and regulative uses of ideas may be regarded as Kant's most original contribution to the theory of hypothesis. A constitutive hypothesis claims to describe how things are; a regulative hypothesis prescribes how inquiry should proceed. Conflating the two produces what Kant calls "transcendental illusion," where reason mistakes its own procedural demands for features of reality. The ideas of pure reason look like hypotheses about the ultimate structure of the world, but they are in fact rules for the systematic organization of knowledge.

Mill will inherit the scientific side of Kant's framework and formalize the hypothetical-deductive method as the standard account of scientific reasoning. The regulative side of the framework, in which hypotheses function as guides to inquiry rather than as descriptions of reality, bears on questions of scientific realism that are discussed also under the ideas of Science and Metaphysics.

Mill gives the hypothesis its most systematic methodological treatment. In A System of Logic, he argues that the "hypothetical method" is a legitimate and indispensable mode of scientific investigation, provided its results are verified by observation. The procedure consists in proposing a cause or law to account for a set of phenomena, deducing what further observations should follow if the hypothesis is correct, and checking whether those observations obtain. If they do, the hypothesis is confirmed, though never with demonstrative certainty. If they do not, it is refuted or modified. Pascal had offered a summary of the logical situation by distinguishing the true, the false, and the doubtful hypothesis: "in order that the hypothesis may be demonstrable, it is not enough that all the phenomena result from it, but rather it is necessary, if there ensues something contrary to a single one of the expected phenomena, that this suffice to establish its falsity."

Mill is clear about the limits of verification. A hypothesis that merely accounts for known facts without predicting new ones has little evidential value, since it might be one of many possible explanations. The real test is novel prediction: the capacity of a hypothesis to anticipate phenomena that were not part of the data from which it was framed. Mill also insists that a hypothesis must be the kind of thing that could in principle be observed or verified, even if present means cannot do so. The question of the task of verification, and of the plurality of hypotheses that may account for the same phenomena, is one that Mill treats with considerable care.

From Bacon, Mill takes the insistence on experimental testing and the elimination of rival explanations. From Newton, the demand that principles be grounded in phenomena. From Hume, the acknowledgment that induction can never yield absolute certainty. Mill's contribution is to show that these commitments are compatible: the hypothetical-deductive method is both rigorous and fallible, disciplined and provisional. The question of how hypotheses are tested, and of the degree of probability they can attain, is treated also under the ideas of Induction and Science.

Mill's formalization of the hypothetical-deductive method became the standard account of scientific reasoning. The question that Plato raised, whether thought can ever transcend its hypotheses to reach unhypothetical first principles, receives in Mill an answer that Plato would not have accepted: that science lives permanently among its hypotheses, refining and replacing them, but never transcending them altogether.