Conjecture

“ The inquiry begins with pondering these phenomena in all their aspects, in the search of some point of view whence the wonder shall be resolved. At length a conjecture arises that furnishes a possible Explanation, by which I mean a syllogism exhibiting the surprising fact as necessarily consequent upon the circumstances of its occurrence together with the truth of the credible conjecture, as premisses. On account of this Explanation, the inquirer is led to regard his conjecture, or hypothesis, with favour. ”

“ The philosopher's stone that should transmute into one another the as yet unanalyzed elements, and produce them from a higher element, if not from primeval matter itself, must be discovered before the first conjecture as to the development of apparently heterogeneous, from actually homogeneous matter, becomes possible. ”

“ They have allowed the word to come into use, even among themselves, for a conjectural explanation, or a speculative conjecture, or a working hypothesis, or a project which has not yet been tested by experiment, or a plausible and harmless theorem about transcendental relations, or about the way in which men will act under certain motives. ”

“ The conjecture of a philosopher so familiar with nature may sometimes be more pregnant with truth than the best established experimental law discovered by empirical inquirers, and though not bound to admit it as a physical truth, we may accept it as a new idea by which our mathematical conceptions may be rendered clearer. ”

“ Now what our modern physicists fail to see is, not only that these two questions are distinct in detail, but that also they are distinct in kind; that a want of power to answer them means, in the two cases, not a distinct thing only, but also an opposite thing; and that our confessed impotence to form any conjecture at all as to the first, does not in the least exonerate us from choosing between conjectures as to the second. ”

“ Everybody liked better to conjecture how the thing was, than simply to know it; for conjecture soon became more confident than knowledge, and had a more liberal allowance for the incompatible. ”

“ But that, as yet, the endeavor to apprehend fundamental Principles, in other spheres than these, has been attended with only partial success; and hence, the ability to establish a Mathematical or Demonstrable basis for other regions of Thought is yet wanting, so far as is commonly known.
When, therefore, we emerge from the domains of Mathematics, Astronomy, and Physics, we are leaving the field of positive assurance, of undeniable truth, and entering the realms where opinion, conjecture, and variable degrees of certainty prevail. ”

“ But besides the indisputable Principles of Geometry, the universal Law of Analogy and Similitude of things, have a Privilege to assist us, in Conjectures relating to the heavenly Bodies, and though not of equal Force with the former, is often as conclusive as the Subject requires. ”

“ The Probability of the foregoing Conjectures, chiefly built upon very distant Observations, shew an apparent Necessity for some other kind of Doctrine permitted by Providence, to give Mankind a Knowledge of their Immortality and Dependance upon it, in the first Ages of the World. ”

“ On the other hand, there must be in physical science an infinite number of conjectures, which can never become certainties; because the phenomena of nature are not given as objects dependent on our conceptions. ”

“ or, if they list to try
Conjecture, he his fabric of the heavens
Hath left to their disputes, perhaps to move
His laughter at their quaint opinions wide
Hereafter; ”

“ OF THE LAW OF UNIVERSAL CAUSATION. § 1. The phenomena of nature exist in two distinct relations to one another; that of simultaneity, and that of succession. Every phenomenon is related, in an uniform manner, to some phenomena that coexist with it, and to some that have preceded and will follow it.
Of the uniformities which exist among synchronous phenomena, the most important, on every account, are the laws of number; and next to them those of space, or, in other words, of extension and figure. The laws of number are common to synchronous and successive phenomena. ”

“ All that is demonstrated in mathematics, and in any other science whatever, is eternal and immutable, since the effect of the demonstration is to show that the thing cannot be otherwise than as it is demonstrated to be. So, in order to understand the nature and the properties of things which I know, for example, a triangle, a square, a circle, or the relations of these figures, and all other figures, to each other, it is not necessary that I should find such in nature, and I may be sure that I have never traced, never seen, any that are perfect. ”

“ Every theorem in geometry is a law of external nature, and might have been ascertained by generalizing from observation and experiment, which in this case resolve themselves into comparison and measurement. But it was found practicable, [Pg 155] and being practicable, was desirable, to deduce these truths by ratiocination from a small number of general laws of nature, the certainty and universality of which are obvious to the most careless observer, and which compose the first principles and ultimate premises of the science. ”

“ Because there may be four intersections of two conic sections, any one of them is not to be found universally, but by an equation of four dimensions, by which they may be all found together. For if those intersections are severally sought, because the law and condition of all is the same, the calculus will be the same in every case, and therefore the conclusion always the same; which must therefore comprehend all those intersections at once within itself, and exhibit them all indifferently. ”

“ But let your positivist attempt to explain transcendental mathematics, and he is all at sea, if he does not borrow from the ideal science or philosophy which he professes to discard. How will the geometrician explain his infinitely extended lines, or lines that may be infinitely extended? A line is made up of a succession of points, and therefore of parts, and nothing which is made up of parts is infinite. The line may be increased or diminished by the addition or subtraction of points, but the infinite cannot be either increased or diminished. ”

“ This fundamental law must resemble the truths of geometry in their most remarkable peculiarity, that of never being, in any instance whatever, defeated or suspended by any change of circumstances.
Now among all those uniformities in the succession of phenomena, which common observation is sufficient to bring to light, there are very few which have any, even apparent, pretension to this rigorous indefeasibility: and of those few, one only has been found capable of completely sustaining it. ”

“ To illustrate, — suppose that a mathematician wants to prove something about the value of π, or about the universal laws of Arithmetic, or about the properties of a continuous function, or about the sum of an infinite series, or about the mathematical relationships of two infinite collections of ideal objects. What he is concerned to demonstrate, lies in the realm of the infinite, and of the eternally valid. And our direct experience gives us only the passing data and the fragmentary ideas of the moment. ”

“ The universality of physical causation, the interpretation of all things in terms of a never-ceasing redistribution of matter and motion, is naturally to them an idea utterly alien. The mathematician, too, and the mathematical physicist, occupied exclusively with the phenomena of number, space and time, or, in dealing with forces, dealing with them in the abstract, carry on their researches in such ways as may, and often do, leave them quite unconscious of the traits exhibited by the general transformations which things, individually and in their totality, undergo. ”

“ Concerning the ultimate modes of existence or action, we know nothing whatever; hence, a law of nature cannot be demonstrated in the sense that a mathematical truth is demonstrated. Yet so great is the constancy of uniform sequence with which phenomena occur in accordance with the laws which we discover, that we have no doubt respecting their validity. ”

“ If we attempted to deal fully with this logic, we should not have time to reach any other aspect of our subject; we will therefore, while acknowledging that it deserves a long discussion, pass by its central doctrines with only such occasional criticism as may serve to exemplify other topics, and concentrate our attention on such matters as its objections to the continuity of motion and the infinity of space and time—objections which have been fully answered by modern mathematicians in a manner constituting an abiding triumph for the method of logical analysis in philosophy. ”

“ As we admit no such propositions, other than as truths inductively collected from observation, even in geometry itself, it can hardly be expected that, in a science of obviously contingent relations, we should acquiesce in a contrary view. Let us take one of these axioms and examine its evidence: for instance, that equal forces perpendicularly applied at the opposite ends of equal arms of a straight lever will balance each other. What but experience, we may ask, in the first place, can possibly inform us that a force so applied will have any tendency to turn the lever on its centre at all? ”

“ Geometrical Co-existence, when it is not a matter of definition (as 'a square is a rectangle with four equal sides') , is deduced from the definitions and axioms: as when it is shown that in triangles the greater side is opposite the greater angle. The deductions of theorems or secondary laws, in Geometry is a type of what is desirable in the Physical Sciences: the demonstration, namely, that all the connections of phenomena, whether successive or co-existent, are consequences of the redistribution of matter and energy according to the principle of Causation. ”

“ Now, what we need for our induction as to the real and absolute efficient of mathematical order and harmony in the motions of the universe is a similar analyzed instance, where the naked antecedent and consequent shall be detected in the very act of conjugation. And, by a propitious arrangement of nature in the great fact of our complex organization, we have it in our power to perform this decisive experiment in the same manner and with as much certainty as in the previous example. We can act as individual causes, either with or without the presence of a rational purpose. ”

“ It is a matter of more than curiosity to consider, to what peculiarity of the physical truths which are the subject of geometry, it is owing that they can all be deduced from so small a number of original premises; why it is that we can set out from only one characteristic property of each kind of phenomenon, and with that and two or three general truths relating to equality, can travel from mark to mark until we obtain a vast body of derivative truths, to all appearance extremely unlike those elementary ones. ”

“ How, being so contained and implied, it could be new truth, and how the theorems of geometry, so different in appearance from the definitions and axioms, could be all contained in these, was a difficulty which no, one, I thought, had sufficiently felt, and which, at all events, no one had succeeded in clearing up. The explanations offered by Whately and others, though they might give a temporary satisfaction, always, in my mind, left a mist still hanging over the subject. ”

“ In our third lecture, for example, we saw that the part of physics which depends upon testimony, and thus upon the existence of other minds than our own, does not seem so certain as the part which depends exclusively upon our own sense-data and the laws of logic. Similarly, it used to be felt that the parts of geometry which depend upon the axiom of parallels have less certainty than the parts which are independent of this premiss. ”

“ Why are mathematical certainty, and the evidence of demonstration, common phrases to express the very highest degree of assurance attainable by reason? Why are mathematics by almost all philosophers, and (by some) even those branches of natural philosophy which, through the medium of mathematics, have been converted into deductive sciences, considered to be independent of the evidence of experience and observation, and characterized as systems of Necessary Truth? ”

“ To assume, for example, that objects which we see continue, after we have turned away our eyes, to be more or less analogous to what they were while we were looking, is a less violent assumption than to assume that such objects are composed of an infinite number of mathematical points. Hence in the physical study of the geometry of physical space, points must not be assumed ab initio as they are in the logical treatment of geometry, but must be constructed as systems composed of data and hypothetical analogues of data. ”

“ For if one asserts, as his account of the nature of Being, that certain ideas of possibilities of experience are valid, he is so far left with a world of objects upon his hands whose only character, so far as he yet defines the Being of these objects, is that these objects are in agreement with his ideas. Such a definition of Being constituted the whole outcome of the Third Conception. The mathematician’s ideas, as present to himself, take the form of observed symbols and diagrams. ”

“ The peculiarity of that species of evidence which is called demonstrative, and which so remarkably distinguishes our mathematical conclusions from those to which we are led in other branches of science, is a fact which must have arrested the attention of every person who possesses the slightest acquaintance with the elements of geometry. ”

“ We have, then, three domains in which the true Deductive Method is active; in which we can start from universally recognized Truths and proceed, by irresistible Inferences, to ulterior Principles and Facts. In three Sciences, in Mathematics as commonly defined and understood, in Astronomy and Physics as herein circumscribed, we are able to establish starting points of thought with Mathematical certainty, and to deduce from them all the Phenomena of their respective realms. ”