# The Philosophical Review/Volume 1/A Mathematical View of the Free-will Question

A MATHEMATICAL VIEW OF THE FREE WILL QUESTION.
BEFORE discussing objections to the doctrine of Freedom, let me try to state what, in my own thought, that doctrine really is. My notion of the matter is more or less analogous to the ordinary theory of Attention. As the current of suggestions, whether determined mechanically or otherwise, passes before the ego, the ego puts forth a free selective power whose result is physically directive. Here Stewart's account of attention may or may not come in: either the ego may merely arrest the current long enough for the mechanism to do the rest, like the "type-wheel" in the telegraph, which prints whenever it stops, or else, possibly, the ego may deflect the current without arresting it. Does the ego here act apart from motives, feeling no pressure from them, or as one among motives, and often against some of the others? i.e. if by "Volition" we understand, not the observable resultant of the ego's free and original act as combined with other causes, but only that act, pure and simple, then are all volitions alike as to strength, intensity, degree of effort, — or do they differ in this respect according to the strength of the opposing motives? Either alternative appears to be tenable; but, in so far as they really differ, I incline to the second one. This raises the question of relative strength of will-power and determined motives, and I incline to believe that the will-power, in its possibilities, and perhaps even as commonly exerted, — is by no means infinitesimal in comparison.
Thus motives and volitions would belong to a system of what we may call "spiritual forces," because like physical forces they may oppose one another, conspire and perhaps combine into intermediately directed resultants; but these spiritual forces cannot so conspire with or oppose the actual physical changes ​or motions of the moment as to "do work," but can only produce deflection or transference of an energy to whose potential they do not contribute. I regard them as quasi-perpendicular to all physical forces: i.e. their action upon any physical atom is, in the above sense, always as if at right angles to its path, so that of course they cannot increase or diminish its kinetic or potential energy, except indirectly by deflecting it into new positions and conditions, where, however, all transfers of energy will still be made under purely physical laws. [Here, and in what follows, we use the term "physical force" only in the sense of a push or pull exerted at a material point by attraction, repulsion, or vis inertiæ; and analogously, the term "spiritual force": so that a group of agencies and phenomena, like electricity or intelligence, would not be called "a force."]
Even if the concept of quasi-perpendicularity, which perhaps has been already suggested by Maxwell and others, should rest upon nothing deeper than a metaphor or convention, yet it is the more presumably natural and helpful here because of its known value in Pure Algebra. Thus, the perfect symmetry between the relations of i (=${\sqrt {-1}}$) to +1 and to -1; or of 1 to +i and -i; i.e. that mutual independence of the two abstract units 1 and i which would seem so analogous to the independence of physical and spiritual forces, — has led to the use of the "complex plane," with resulting methods of great power and beauty. On this plane every value of A+Bi is located at a point having longitude A and latitude B, as in the annexed scheme; so that i is treated as quasi-perpendicular to 1.
| -1+i | i | 1+i |
| -1 | 0 | 1 |
| -1-i | -i | 1-i |
In mathematics, this notion of quasi-perpendicularity, though as I think often implied, goes unnamed. It is very flexible: a translation and a rotation would be quasi-perpendicular, however their respective directions were related, and so would be the several parameters of a curve when they were regarded as a new system of variables. Let us hold the concept in a correspondingly free way. Let it simply accentuate for us the fact that ​the ego does no physical work, but can only by hypothesis decree something as to the direction of work, while it may or may not connote for us anything that is geometrically more definite.
This quasi-perpendicularity appears to me the natural, and perhaps almost necessary, solution of other difficulties that have nothing to do with freedom, but only with the apparent dualism of mind and matter, and with their apparent interaction; while it may also sufficiently suggest the deeper unity in which that dualism doubtless rests. We do somehow receive impressions from the outer material world, and do, whether freely or as mere automata or channels of influence, produce impressions upon it; while, on the other hand, we are aware of phenomena in ourselves, like intelligence, joy, gratitude, obligation, remorse, which seem to us essentially independent of space and matter, though they often have reference to the external world. Now I know of no other figure by which, so well as by that of quasi-perpendicularity, we can represent to ourselves this apparent independence and this apparent interdependence of the physical and the psychical. Our hypothesis may be held as hardly more than such a figure, and it still suffices for the present purpose: though I incline to think it is not merely a figure of speech.
Admitting in a general way such a solution, many curious questions of detail would remain which we cannot as yet adequately discuss; nor need we, for the purpose of the present argument. For instance, shall we think of the quasi-perpendicularity as mutual; the physical forces exerting a merely directive effect upon the spiritual phenomena, as well as the spiritual forces upon the physical phenomena, and the law of conservation presumably holding in the spiritual as well as in the material system? Is there any spiritual force that involves the element of Time, or Inertia, and in virtue of which every system of spiritual forces must be in equilibrio just as a falling stone hangs balanced between the pull of gravity and its own resistance to further acceleration? Again, must this quasi-perpendicularity be regarded as a mere convenient metaphor, or may it be an actual perpendicularity, the spiritual forces pulling in a space of their own, more or less like our known space? In the ​latter case, are the two spaces coincident; or does the spiritual include the other as that includes a given plane or line; or is every line and every plane of one space perpendicular to every line and plane of the other? The third alternative seems untenable; for, admitting it, how could a spiritual force produce even a deflective effect in physical space? But the other two remain, and the second may fall in with such purely physical speculations as W. W. Rouse Ball's, who seeks to obtain both Newton's law and certain results of spectrum analysis from the virtual hypothesis that our known space, lying like a plane in a larger space, can vibrate in the fourth direction. [Messenger of Mathematics, June, 1891.]
It remains now to notice briefly, from the standpoint of mathematics, certain supposed difficulties of the Free Will hypothesis. We are told that the Determinist theory is forced upon us:
(1) By the principle of conservation of energy;
(2) By the law of causation;
(3) By statistics and the known effect of environment upon conduct and character; and even
(4) By consciousness.
(1) The first difficulty disappears with the admission of quasi-perpendicularity; and, as I have tried to show, some such admission is almost necessary in order to explain other phenomena than those of our supposed freedom.
(2) If this means that all causation is physical, in the sense that excludes independent contributions to the stream, then it simply begs the question. If it means that an infinite mind, knowing perfectly well the present, could with certainty trace thence the stream of events both upwards and downwards, this again is an assumption, made plausible by hereditary ideas as to Fate, and the Divine Decrees and Foreknowledge, and perhaps also by two fallacies as to the fixity of the Past. Doubtless the Past is fixed, if only because it is past; but since the same fixity would result if the Past were wholly inferrible from the ​Present, one may illogically deduce the inferribleness from the fixity and assign it as a cause of the fixity. And again, it is natural to assume a symmetry and reversibleness of relation between Past and Future which does not really exist, and thence to conclude that if one be fixed, the other must be so too. This argument is not that "from Causation," yet one seems to suggest the other. Now, however it be to the philosopher, Past and Future are not alike to the scientist. Even if physical causation be considered as reversible, many of the cycles of actual change are not. These cycles move in fixed directions, and not backward, even when they bring events and successive individuals or systems around to nearly the old conditions. And besides, there are changes going on which, as far as science can yet see, never will be unmade or offset by any cycle: for instance, the progressive residual concentration of mass and diffusion of heat, and the unceasing Loss of "motivity." Thus neither from moment to moment nor yet in the long run is the relation of Past to Future a reversible one.
But if the objection merely means this, "Since every effect or phenomenon has a cause, and this cause has a cause, etc., thus giving an infinite number of steps of causation between the effect and its first cause, therefore free volition cannot be a cause, for this infinite number of steps between it and its observed effect must take an infinite time," — if this be the supposed difficulty, then I think it turns upon the same fallacy as does the story of Achilles and the tortoise. We are told that he could not catch the tortoise, because while he ran 100 rods it crawled one; and while he ran that one it crawled 1/100; and then 1/10000 and so on. Thus analyzed, the process did indeed require an infinite number of steps, but then these steps grew so rapidly shorter and shorter that the sum of all their lengths was finite, and might have been microscopic. Very likely it is as true in our problem as in that of Achilles, that one may rightly either introduce the infinite series or not: i.e. we may say "Everything is caused," and so bring in the infinite series, or say "Volition is its own cause" and stop there, replacing the series of steps by its resultant or quasi-sum.
​(3) Statistics do indeed show that, of a given people under given circumstances, one in so many misdirects a letter, or commits a murder, or can read three languages ; and that the ratios remain very nearly constant from decade to decade unless the environment changes. But this is precisely what would happen if human actions were the resultants of environment and of a factor of spontaneity which, like a throw of honest dice, was absolutely unpredictable. We should get, just as we now do get, agreements where the basis of estimation is large, and discrepancies where it is small; indeed, if the argument from statistics were worth anything, then we could prove in like manner that the throws of dice are governed by discoverable laws and not by chance, — which is practically untrue.
So with character: if it be the joint product of environment, heredity, and volition, then in the long run it may seem to result mainly or wholly from environment and heredity, for the reason that volition (unless guided by principles whose development is, itself, largely due to environment, etc.)] is like dice throwing, now this way and now that, so that its net result varies nearly as the square root of the number of cases that have come up, while the result of a steady pull from heredity or environment varies as the number of cases.
(4) To examine the facts of consciousness belongs more to the philosopher than to the mathematician. My own consciousness, however, appears on the whole to testify neither to complete determinism nor to complete freedom; and I suspect that its evidence is really in close accord with the theory sketched out above.
It would seem then that the analogies of physical and mathematical science are not unfriendly to that old faith in freedom to which the conscience and common sense of the race have substantially held; and if we cannot now hold to it in the old simple way, this is mainly because the world as known to us is so much larger and more complex than as known to our fathers. We cannot do or become at once just what we would: the ego can act effectively only through, or as part of, an intricate mechanism, and perhaps much that it seems to initiate is mere ​ideomotor reaction. Yet I think its directive agency does tell in making results to be somewhat other than they would have been, and thus gradually moulding character and environment, and the heredity of those who shall come after. Moreover, the effect upon the inner character may be more rapid than upon the apparent character and the environment: for if there be a certain element of earnestness, making the inner results fairly constant as to direction, these results should tend to accumulate proportionally to the number of occasions, while the effects of environment upon the inner character would be more conflicting and so would follow more nearly the law of the square root. Thus, though development has taken the place of sudden creation in our new world, it may still be that we do, little by little, create our own characters, and not merely the character of the race.
J. E. Oliver.
Cornell University.