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The "Phenomenological" Base of Physics

Henry Flynt

© Henry A. Flynt, Jr.


[1999. I endorse this essay’s identification of the vulnerable junctures in physical doctrine. The question arises whether to try to sharpen the arguments on the basis of more detailed knowledge of the junctures in thermodynamics, probability, metamathematics, etc. cited here. Perhaps what is needed is not so much to sharpen this essay as to find readers who can spring from these hints to the next phase of supersession.]


Section I

A. My aim is to escape from the scientific way of apprehending the inanimate world. This enterprise anticipates a culture which annuls physical reality–reaching all the way to physical reality's base in common sense. Thus, the audience for this manuscript consists of those who wonder whether physics could be even slightly untrue–or whether, indeed, it could be wholly false.

I choose physics as the topic here because it is the fundamental branch of the quantitative rationalization of nature–in the research program stemming from ancient Greece.

B. One could write interminably to define the scientific project as it stems from ancient Greece. (Cf. Karl Popper, Quantum Theory and the Schism in Physics, pp. 162-4.) For my purposes, the following features are notable.

1. The only portion of human understanding which carries authentic evidence or impressions of the objective world is alert waking consciousness.

2. Mind is one of a human's organs-it is an organ characterizing humans.

3. Time is the seriality of world-events.

4. The reality-type of the world's constituents is idealized thinghood or thing-like abstraction. A method of artificial stringency regarding meanings is posited: logic. Through logic, sensuous-concrete things are found to be subtended by ideal, abstract things. The latter nevertheless remain "things" in being self-contained, inert, insightless.

C. The scientific enterprise finds quantitative regularities in the inanimate world by viewing the inanimate world in the perspective of such principles as the following. The principles are given in their early forms.

1. Equilibrium. A system tends to stasis or repetitive motion. Also: what are the conditions for opposing influences to cancel?

2. Sufficient reason. If an initial state is symmetrical, it cannot devolve/resolve in a biased way unless an external influence is imposed.

3. Symmetry. If conditions which possess a certain symmetry determine an effect (uniquely), the effect will exhibit the same symmetry. (Same as Sufficient Reason.)

Wherever nature seems to possess direction or imbalance, a true law of nature will nullify that semblance. In fact, directedness in nature is only conventionalistic.

4. Conservation. The universe contains a constant total of motion, which was established at the creation.

5. Least action. When a change occurs in nature, the amount of

time elapsed,

distance travelled,

or ...

is the smallest possible.

Much later in this manuscript, I will review some elementary physical regularities discerned via these principles: the lever with equal weights; the unison monochord; the angle of a ball bouncing off a backstop; the pendulum; etc. But my decision to emphasize accessible examples which require little or no instrumentation must not lead us to misjudge the foregoing principles. We cannot suppose that the principles were abstracted directly from one or another pragmatic problem. Rather, the principles are theological theses adverting to the cosmos as a unity. Moreover, it is quite possible that the precedent for these notions was economic. (Market-clearing price; primary-input constraint; minimization of cost.)

... science can accomplish nothing by the consideration of individual facts; from time to time it must cast its glance at the world as a whole. Galileo's laws of falling bodies, ... nay, even the concept of mass, could not, as we saw, be obtained, except by the alternate consideration of individual facts and of nature as a totality. ... purely ... elementary laws do not exist. It would be illogical, accordingly, to exclude as less certain this necessary view of the All, or of the more general properties of nature, from our studies.

A great many natural phenomena ... produce the impression of economy, simply because they visibly appear only when by accident an economical accumulation of effects takes place. ... We facilitate instinctively our comprehension of nature by applying to it the economical ideas with which we are familiar.

Ernst Mach, The Science of Mechanics, p. 461, pp. 459-60

But the principles go beyond the immediate in a much more serious respect than addressing things as a whole. Before the principles can have application, the phenomena of the environs must already have been reified and idealized.

a. Objects must have been abstracted from their sensuous mutability. They must have been divorced from mutability of their visual size. They must have been divorced from mutability of their visual location–acquiring positions in objective space.

b. Substantiality must be imputed even though it is not given in sensation. As I look at two bodies which are piled vertically, I must conclude that the lower body bears the load of the higher body.

And yet physics paradoxically negates this common-sense tenet by working with dimensionless and weightless bodies.

c. A single event must be able to exemplify a vast and significant family of phenomena.

D. Let me turn from self-definitions of the physical enterprise to the repressed question of the relation of physics to common sense. What I am concerned with in this manuscript is definitions of time, location, weight, materials, etc. which can be implemented by humans without the mediation of instrumentation–and which make it possible to begin the physical enterprise. We have seen that physics presupposes experiences with objectified, substantial, exemplary phenomena. Time is conceived as one-directional. Time and space are conceived as homogeneous. Weight ratios are defined as ratios of geometric sizes of bodies made of the same material. The preferred mathematics involves traditional positive-integer arithmetic and Euclidian geometry.

Early atomic theory was a speculation that the world might be discrete on an imperceptible level. "Atom" actually means a simple. But then atomic theory was already in contradiction with the preferred mathematics. Theoretical difficulties in positing imperceptible irreducible atoms. Collision between two atoms would be totally inelastic: instantaneous acceleration; infinite force.

Early atomic theory already explains the palpable world via speculations about entities and actions which are far below visibility. So, from the point of view of this manuscript, it already belongs to a derivative phase of the scientific project. For that reason, it is already outside the scope of this study.

Still less, then, will this study consider the derivative speculations of twentieth-century physics: two-dimensional time; etc. The "high" end of physics will be noted here only when it trips over questions of principle that reflect on the existence of physics at all.

1. The notion that laws of nature existed before there was anything at all.

2. The "conjugacy" of entropy and information. (Does this make information a (linguistic-cognitive) substance?) Cf. "Entropy and Information" in Paul Chambadal, Paradoxes of Physics, p. 96ff.

3. The notion that nature's lawfulness is eroding itself via entropy.

E. Scientists and positivists seem to get great pleasure from disparaging ordinary language and common sense. Nevertheless, common sense remains an operational and "phenomenological" prerequisite or base of physics. We began to see this with I.C.a and I.C.b.

1. Experimentation and technology must begin and end in the life-world.

2. When one uses propositions about levers, point masses, etc., applying theoretical techniques to practical problems, one returns to the common-sense realm.

3. The notions which give intuitive meaning to physical theory come from common sense: apprehending levers and point masses as material objects.

4. When you drop an object, it may in falling become invisible. Before physical abstraction begins, you invoke common-sense substantiality to be sure that the body is not an apparition. In fact, the body must qualitatively transcend any such thing as a correlated stream of apparitions belonging to sight, touch, hearing.

5. Common sense is posited as a survival competency for the child.

a. Distinct, enduring, and recurring objects.

b. The reality of classes of things, allowing common names.

c. Size-constancy and shape-constancy.

d. Things which remain the same when change occurs.

e. Things which remain the same when process occurs.

f. Regular repetition.

g. Cause and effect.

A work by Noam Chomsky, Reflections on Language (1975), is notable here for the place it assigns to common sense in thought. For Chomsky, common sense is to physics as ordinary language is to the mathematical "language" of physics. Just as Chomsky finally had to explain the innateness of universal grammar by positing that it is congenitally hard-wired in the brain, he posits also that common-sense notions are hard-wired in the brain. Then, Craig Dilworth lists, as principles of classical science, the common-sense points which I have just listed as (a)-(g).

6. Physical notions cannot be elementary. Physics cannot compete with common sense as a medium of everyday apprehension of the world and everyday interaction. The simplest physics already invokes the infinitely small, the infinitely large, ideal or "divine" geometry, point masses, action at a distance, etc.

Aristotle merely formulated the most commonplace experiences in the matter of motion as universal scientific propositions, whereas classical mechanics, with its principle of inertia and its proportionality of force and acceleration, makes assertions which not only are never confirmed by everyday experience, but whose direct experimental verification is fundamentally impossible: one cannot indeed introduce a material point all by itself into an infinite void and then cause a force that is constant in direction and magnitude to act on it; it is not even possible to attach any rational meaning to this formulation. And of all the experiments by means of which textbooks of mechanics are wont to prove the fundamental law of dynamics not a single one has ever been carried out in practice. Aristotelian physics thus has the advantage over classical mechanics in that it deals with concrete, observable situations constantly encountered. But from a scientific point of view this very advantage constitutes its weakness, for these situations are so complicated (the reader need only think of a vehicle drawn through the air along a rough road, or of a body of any given form thrown upwards) that even with the aid of perfected classical mechanics they can be treated mathematically only by approximation and at the expense of comparatively arbitrary suppositions. The theory of motion calls for just as extreme an idealization as that by means of which Euclidian geometry is deduced from physical experiences of solid bodies. [italics added]

E.J. Dijksterhuis, The Mechanization of the World Picture, pp. 30-31

Let me attempt to restate my point more incisively. The picture of a lever employed in physics cannot be related to the sensuous-concrete world, since it comprises an immaterial lever supporting material loads. (Indeed: by exploiting this deliberately, without continually curbing it via common-sense intuition, we might elaborate a new science.) The physical lever is an idealization of the reified lever provided by common sense. The purpose of the physical idealization is to eliminate the weight of the lever arms as a factor in the equilibrium calculation. Without the reified lever of common sense as the precedent and as the practical concern, the physical idealization would not have been propounded.

The point mass is not anything in the world. But you are permitted to refer this abstraction to some things in the world and not to others. A wisp of smoke can have or be a point mass, a shadow cannot. Can a light ray be or have a point mass?

The only way one can ever apply point masses and weightless levers to practical problems is to understand them as abstractions from common-sense experiences with objecthood. The constituent of perception or lived experience called the object-gestalt. (Perception integrating visual and tactile "aspects of an" object to yield "the" object.)

F. Another aspect of physics' repressed dependence on the "life-world" is that it implicitly posits "human minds" even though it excludes them from its ontology.

1. Physical science–as represented by neuroscience–seeks to explain "the realm of thoughts" away by reducing it to the realm of brain-states. But preponderantly, science consists in reducing nature to "appreciated propositions," understood propositions: mapping the realm of physical states to the realm of meanings (?). Prediction, confirmation; choice, action.

2. As one watches a drop of ink spread in a container of water, the IMPERSONAL qualities of this event IN ONE'S EXPERIENCE do not establish that the process runs from past to future. Only when one's memory, anticipation, and willing can be invoked as evidence can one say that dispersion of the ink-drop is future-ward.

3. Without free choice, the concept of an experiment would be vacuous. Physics would be the passive observation of the externally given, as astronomy is. The physicist must be conceived as facing a "bundle" of possible event-chains, one of which is realized because he chooses it. The other event-chains in the bundle are possible but not actual. The bundle is constrained: not every thinkable chain of events is possible.

Moreover, to explicate further the notion of an experiment, it must be the mind which makes the selection of one from many possible event-chains with those objects in that location. In this sense, the mind makes the external event which it studies. In this non-trivial sense, experimentation is already telekinesis.

G. Let me briefly comment on how experiential time is objectified–before being abstracted to objective time. Subjective features which do not determine the objective world are strained out. Nothing in "the world" depends on which event is taken as zero. Nothing in "the world" depends on how a given series of discrete events are numbered so long as the numbering increases monotonically. (This then generalizes time to before my birth and after my death.) Next, add to the time-scale all possible numerical interpolations, thus making a place for events which intrude on one's awareness, for events which one does not intend or attend to. (The time-scale then has labels for more moments than I can specifically attend to or plan events for. "The world" has more events than my acts of attention determine.) In stereotyped applied mathematics, interpolation in the time-scale is carried on until the real-number continuum is reached.

But this explanation slides over major issues. Whether successive durations can be made comparable or congruent. From ordering to a metric. Also, whether the meaningfulness of irrational ratios in the tangible world has been established.

The result of the conceptualization is a subjective time which is like an (infinitely subdivided) ruler–except that its orientation is fixed (from past to future).

A phase which had to be established prior to all this. The ability to recognize events–which involves judgments, based on memory, that events have subsequent copies (events are repeated).

H. Physics has been shaped by Euclidian geometry and by the algebra of quadratics, etc. Thus, it analyzes physical problems in real-number space, as we just saw in the case of the time-scale. This then poses the question of whether nature actually supports irrational-ratio relationships.

But there is already a crucial mediation which I will not address until Section II. Physics is a theory of the idealization; and the empirical phenomenon is always an inferior and problematic adjunct. The ratio between the circumference and diameter of a circle is . To ask whether this is empirically true is to take a situation with multiple sources of inexactness too seriously. Moreover, the ancient Greeks did not possess our refined picture of the microcosm. They could not know that at a small scale, a visible drawing of a straight line becomes an incomparably different configuration (a subatomic configuration). But that doesn't obviate the question, because subatomic processes may be claimed to be governed by irrational ratios.

Ancient physics has cases in which irrational-ratio relationships are established in the physical idealizations. The lever with irrational-ratio weights and arms (analyzed by Archimedes). The tritone monochord, which is fully allowed by the premises of ancient physics; but which I was the first person to design and build.

Pure geometry and sufficient reason assure that physical magnitudes occur in irrational ratios.

But in modern physics, the issue becomes hopelessly obscured by a new methodology. A priori magnitudes are confounded in theoretical analysis with magnitudes which exist only as empirical measurements: acceleration of gravity; Young's modulus.

The period of a pendulum of small arc is proportional to 2 times the square root of its length, suggesting that period and length could have an irrational ratio. (Then one could establish irrational time-intervals.) But the constant of proportionality is g[exp -1/2]; and g is defined only by measurement, i.e. to so many significant decimals. (In any case, the analysis of the pendulum as simple harmonic motion is a falsifying idealization, for both geometric and thermodynamic reasons.)

Real analysis is trusted in modern physics with a religious fervor. And yet this methodological incongruity renders the seeming exactness of the derivations nonsensical. The classical theoretical physicists, it appears, must have worked in a state of dichotomous delirium.

It is of interest that ancient physics was able to analyze levers and pitch intervals without invoking empirical constants. They did this even though, according to modern thinkers, the static moment can only be discovered through observation. The purely geometric proofs of the static moment given by the ancients are now regarded as circular.


Section II

A. In I.F.3 above, I noted that a scientific experiment is an actualization relative to possibilities which remain unrealized. I now turn to the main topic of this manuscript. What does it mean to say that nonactual possibilities exist? What does it mean to say that nonactual possibilities are influential? When we talk about nonactual possibilities as being significant, how do we lift "reality" off from the actual historical career?

For contrast, let me analyze some situations in lived experience which has not been physically idealized. When watching a suspense movie, we may subjectively empathize when the movie depicts a hazard, or depicts a person faced with choices. Objectively the future of the movie is pre-embodied in celluloid. So what is our experience of possibility? We are ignorant of the pre-embodied future. But don't we also impute a context of possibilities, when actually there is no bundle of possibilities whatever?

In a dream, I can be in a hazard situation. Is this sense of hazard objectively meaningful?–or is it meaningless? One theory of dreaming is that I make it up as it goes along-the future is open in the sense that what thoughts I will have in the future is not pre-decided. If in a dream I am threatened by a hazard, waking thought doesn't conceive it as an autonomous physical hazard. Rather, whether my mind shall subject me to a nightmare. The reason why there is a bundle of possibilities is different from the reason in physics. It's not that my choice interacts with a physical order, but that my thought process has an open future. If the action in my dream is indeed my mental invention, then the issue of an unknown but inevitable future is an issue of whether my conscious life is hostage to a latent unconscious thought-process which is involuntary. In the sense that there is no autonomous physical world in the dream, there are no possibilities outside me which remain genuine even though they are not actualized.

Sometimes I have a dream in which something is progressively happening; I expect that I know what is going to happen next; I direct my attention more closely; but the happening veers away from what I expected. The expected thing involved disturbing self-disclosure. This suggests that on a level less conscious than my patent consciousness in the dream, I re-script the dream while it proceeds. This suggests in turn that at a hidden level, I improvise the dream (as it proceeds). Invoking the notion of a possible event-chain in mental life, where wish, unease, and mentation hidden from myself are considerations. Ascribing possibility to an anticipated but unused step in an improvisation whose operation is hidden from me. It was possible that the dream's story would go from x to y, but it actually went from x to z. Does this give a glimpse of a hidden moment-to-moment dream-inventing process, along with self-censorship, which in turn would require disquiets which I may not acknowledge as such consciously?

B. Returning from lived experience to physics, I hold that physics is vigorously committed to the reality of possibility. The explication of physics which follows is in my own words, but I avow that it portrays the physical method authentically.

The subject-matter of physics is not actual events but a universe of ideal events. Every actual event is an instantiation of an ideal event; and, moreover, it is an inaccurate or damaged instantiation. The theoretical event has to be conceived as more real, deeper, than the sensuous-concrete event. "A body falling in air is a body trying to fall in a vacuum and failing to do so." Falling in a vacuum shows best the partitioned structure of Nature's order. It best decomposes the event into the naked process or principles. No two actual events are perfect copies. This is a fault which physics has to overcome.

(Physics is not empirical curve-fitting. Physicists know what the elementary and clearest processes are, e.g. inertia.)

The problem of how a mathematical theory can be true and yet not apply to the real world therefore has subtle qualifications. The physicist's real world is a quantified idealization separated from the "historically actual" world.

What, then, of actualization? Once an ideal event is admitted to the ontology, it may be actualized anywhere. That is, it may be initiated anywhere. Once an event is initiated, if it is treated as an isolated system, it then proceeds on a historical course. A body, once released in free fall, cannot be released again without intervention to return it to its original point of suspension; and to so return it cannot be done instantaneously. (Springing the same mousetrap cannot be repeated faster than you can reset it.)

Let me mention some postulates regarding the "dimensions." (Cf. I.G.) Space and time are containers distinct from their contents. Space and time do not influence their contents differently at different locations. The scales, grids, for space and time have names, labels, for more points than I can specifically attend to or produce events at.

If events are instantiations of a single ideal event, then they can literally be repeated. This is so even though time is not claimed to be retraversible or to have branches. Events can be copied ad lib–except as they become mutually exclusive, like trying to put three 1-inch cubes in a box of volume 2 cubic inches at the same time. But you can prove that any of the cubes can be put in the box by copying the box.

C. Before me are two toothpicks, call them left and right. I snap the right toothpick. I could have snapped the left toothpick–but I didn't.

If somebody wants to be so pedestrian that he says that only the actual breaking is real, and the possible breaking is literally meaningless, physics must deny that. There can be no physics in a world which is narrowed to the actual historical career. There would be no experiments; events could not be repeated. Tension and metastability would have no meaning.

What explanation does physics provide that I could have snapped the left toothpick instead of the right? I can't make a replay with the broken toothpick. What I might have done in the first place would have been to snap both toothpicks at the same time. That is not exactly what is to be shown, but it would lend credence that I could have snapped the left toothpick alone. The perfect proof that I could have snapped the left toothpick alone would be for somebody else to use copies of my toothpicks and to snap his left toothpick alone when I snap my right toothpick alone. The superiority of this demonstration is that it realizes the alternative in the same time-slot as the original, thus showing that the original event cannot uniquely claim that time-slot. (Physics' task here, in fact, is to show that it can escape history, whose events do uniquely claim their time-slots. April 14, 1865, for example, is uniquely claimed by Booth's shooting of Lincoln.)

Less ideally, I could use copies of the toothpicks and perform an event in the future of the original (first) event in which I snap the left toothpick alone.

Physics must hope that possibility can be demonstrated by making a copy–ideally, a concurrent copy–which has the claimed variations.

Suppose you wished to analyze the shooting of President Lincoln as a physical event. You would have to have copies of President Lincoln which could be shot mortally or not shot, over and over and over. Ideally, all of these trials should occur in the same minutes on April 14, 1865. That, then, would show that the mortal shooting of Lincoln did not uniquely claim that time-slot. Physics must nullify the dominion of the actual (and of the historical). It must nullify any notion that the way a given time-slot is filled is unique.

In I.D.1, I mentioned the current cosmogeny which goes back before the existence of anything whatever. Here, indeed, is a case in which one of the most rarified topics runs against an elementary, pre-physical issue which can be illustrated with toothpicks. Even the start of the universe cannot be a historical event, an individual event. It must be a product of laws concerning classes of events, an instantiation of a generality.

D. Positivist apologies for physics are a hoax. The event, "snapping of a toothpick," cannot be a mere shorthand for events which occur successively and which are united as "this event" by a gratuitous stipulation, by a conventionalism. One could collate successive sunrises as "this event"; and one could collate successive sunrises and appearances of the Morning Star as "this event." Physics cannot accept that we have "an event" with equal validity in both cases. Let us see why.

1. From the point of view of physics, actual or individual events are idiosyncratic: "inaccurate" or "damaged." Putting several successive actual events together in a class does not help. There could be no talk of a falling apple obeying Newton's laws unless the theoretical event was viewed as more real, deeper, than the sensuous-concrete event, which never satisfies the assumptions (point mass, no friction, no air, etc.). Successive sunrises fail to be exact copies; but physics needs exact copies for its subject-matter.

And then there is a difficulty with actual events from another direction. Refined perception of a sensuous-concrete event always finds detail, subtlety, departure from the physicist's absolute model. So it was that when Galileo dropped his two weights from the Leaning Tower, the Aristotelians claimed vindication because the heavier weight hit the ground an instant before the lighter one. The actual event is the physicist's enemy. This point has been underlined by Christer Hennix's observations on refined perception.

2. Let me turn specifically to why the sunrise/Morning Star "event" is unacceptable. Physics needs successive events whose unity is substantive–not solely the product of stipulation. For there to be physics, there must be unity and disjointness in the world-not merely disjoint classes. Substantive unity of events (such as sunrises) may be logically interrelated to causation.

But in the September 1962 Physics Today, Jerome Rothstein seems to support a phenomenological standpoint in his construction of physics to the extent of affirming that my successive encounters of a thing cannot be united because subjective time does not repeat-even if it is freed from an absolute beginning and made infinitely subdivisible. So it is apparitional similarity (total apparitional similarity?) which determines successive events as repetitions of "an event." The point, in fact, is that this is inadequate for physics.

E. It is claimed that a physical law such as Boyle's law has two different realities: as a formula; subsisting in the world. If something like a proposition can have an unverbalized being-in-the-world, this would undermine the entire modernist doctrine which says that sentences are not names. Sentences would be something like names. Boyle's law would be something like a name of the unverbalized-systematic-in-the-world represented by pumping up an automobile tire.

This notion of a law-subsisting-in-the-world is to be a notion of regularity-or better, of "systematic"-which is independent of causality. Like spontaneous regularity, or regularity without inter-production.

Coincidental correlation: the moon cycle and menstruation.

No event-in-the-world self-evidently has the form of a law.

As we saw with apparitional similarity, physics cannot rest content with coincidental correlation. Coincidental correlation's legitimate role is in defining a metric. Otherwise, if concomitant events have no causal relation, the physicist may have to judge the concomitance to be superficial or misleading.

To the sort of positivist who thinks that the notion of Nature following rules is unavoidably anthropomorphic, then a claim of rules subsisting autonomously in Nature must be an embarrassment.

F. Let me resume my discussion of nonactual possibility from II.C. Let us consider nonactual possibility in physics which is not the discard of a sentient choice. That is to say, let us consider metastability; also the common-sense notion of tenseness or hazard. Something whose behavior is completely inert nevertheless has the aura of sudden violent change.

One scheme for conceptualizing a hazard is probability. A wall which has a 1.0 per cent probability of collapsing is a hazard. Actually the term for a hazard conceived probabilistically is "risk." But to introduce the notion of the fractional likelihood of an event would only create a more difficult task of justification than we have seen already. Early opponents of probability theory argued that its fundamental principle–assigning fractional likelihood to an event on the basis of counting symmetrical outcomes–is specious. To speak of a fractional likelihood of Lincoln's being mortally shot on April 14, 1865 is Just nonsense, it is argued: actual events happen with 100 per cent likelihood, nonactual events have no likelihood.

I shall persist with my interpretation of physics as a doctrine of ideal events. In that perspective, you prove that a pile of bricks is metastable by making an exact copy of the pile, disturbing the copy, and observing that it abruptly collapses. Then you can say that the original pile is metastable even though its behavior remains inert.

Physics must assert that metastable and stable systems are importantly different even when their behavior, their actual historical career, is inert in both cases.

The question of whether there are genuine hazards is real for common sense. A pool of lighter fluid on the floor has an aura of danger even if it evaporates completely without combusting. You should think of it differently from a pool of water even if it doesn't behave differently.

Can hazards reinforce each other? If people smoke in a room, this creates a small hazard of a fire. If people smoke and there is a glass bottle of lighter fluid on a shaky pedestal, then the hazard of fire is much greater. This even if no fire occurs!

Suppose there are on the table a toothpick, and a mousetrap which has been set. I can choose to pick up one or the other and handle it. Now choice is conjoined with a hazardous alternative to yield a compound case of (nonactual) possibility.

Suppose people smoke in a room with a pool of lighter fluid on the floor. Suppose they continue smoking until the fluid evaporates: no conflagration occurs. Yet the hazard level was higher while there was lighter fluid on the floor. If the aim is to make somebody fearful, has nonactual possibility been the threat?

G. Objectives of the physicist are i) knowledge and ii) control. Control is indistinguishable from experimentation, only with the intention of forcing the predicted outcome.

In this connection, our analysis has to consider that the repertory of ideal events in physics is not ordained by a deity, but is dictated by convenience and fashion in the career of physical thought. Physics: sliding a wood block on the floor is not a deep event. Why? Because it mixes motion with a dissipative process. Thus it mars the perfection of the law of inertia, the action principle, etc. It mischievously imposes a unique direction on a prolonged event (motion metamorphosing to heat).

A quandary in the determination of the ideal protagonist or element. If atoms are assumed inelastic, then contact action is impossible.

Having discovered this ideal realm which necessarily is physics' subject-matter, let me continue with some questions about this world which is with us always but which we never apprehend sensuously or concretely.

1. Do the elements of the ideal realm have anything to do with the elements apprehended in common sense (the elements which Chomsky says we are innately wired to apprehend–the preparation which gets activated in childhood)? No, they are not at all the same elements. The child's common experience of sliding a block on the floor cannot be an ideal event, because it is not clean. It illicitly mixes mechanics and thermodynamics, thereby violating the action principle.

On the other hand, I must note as per 1.E that common sense is a source of intuitions which guide the employment of the ideal elements.

2. Is there a chain of being in the realm of ideal events, such that frictionless motion is more divine than motion with friction (since the former is theoretically clearer, more satisfying)?

3. Is the realm of ideal events only an inventory of events, or is it an inventory of event-series expressing physical laws? If you drop an ideal ball in the ideal realm, does it fall to the ideal floor? A physicist would have to say yes, the ideal realm encompasses the event-series which express physical laws. But then the ideal series are ideal histories. Physics is writing histories outside the actual world. There must be a mortal shooting of President Lincoln in the divine realm which perfectly manifests the physics of combustion, projectiles, etc.

4. The terrestrial realm and the divine realm are mixed to different degrees in theorizing. Statics of the lever: the loads have volume and weight; the lever has neither and so is immaterial. If acceleration of gravity is invoked, this is a magnitude which only arises empirically.

5. The ideal realm is more detailed than the terrestrial realm in that the mathematical techniques presuppose infinite detail. But the ideal realm is less detailed in that it strips "repeated" events of their idiosyncracies, their "imprecision." Also the physical model forgets sensuous qualities, and they cannot be recovered from it.

6. Perhaps amplifying (3) above, consider the classical proposition that processes are time-reversible. This would mean that events possess their serial relationship in the ideal realm.

Referring back to II.D and to the beginning of (G), physics must deny that classes of identical events are created by gratuitous stipulation. On the other hand, ingenuous inspection of the world cannot determine what should be an ideal event.

a. One of physics' favorite elements, the point mass, has no sensuous or common-sense existence.

b. Do sugar and salt belong in the same class because they look and feel the same? The answer is that indistinguishability in a single sensory modality is utterly superficial.

c. In what theoretical perspective do you judge ice, water, and steam to be the same substance; and sugar and salt to be different substances? Are you supposed to know this by instinct?

d. When "a" shadow is made to "move" by switching lights, is it the same shadow?

e. A wood block sliding on the floor is a bad event.

f. Inelastic atoms are discovered to be no good when contact action is analyzed by classical mechanics. So genuine simples are no good.

g. Approved ideal elements/events turn out to be no good upon a change of scale. Boltzmann's statistical mechanics turns out at the subatomic level to be the wrong reality.

H. Let me make some supplementary observations on the foregoing.

Nonactual possibility is like the infinitely thin weightless lever. Phantoms support and guide material, actual phenomena. I suggest trying to construct a situation in which this feature is explicit and unrestrained.

Given the choice of a toothpick and a set mousetrap, suppose I handle the mousetrap and it doesn't spring. I incurred a risk, and I could have chosen a safe alternative. But the risk never "materialized."

I suggest trying to construct a situation with a second degree of nonactuality.

Can a nonactual possibility be a rational object of fear? Cognitive-emotional involvement with phantom events.

Physics propounds entire phantom histories. Moreover, it seems that these shape actual history. I.e. what happened to Lincoln devolved from variations involving ideal ballistics, etc.

Classical mechanics says that if you saw a film of April 14, 1865 in which the bullet jumped from Lincoln's body to Booth's gun, you could not infer that the film was being run backward. Physics never disavows this position. It achieves realism by gratuitously switching to a different branch, the science of heat.

I. For me, the earliest revelation of physics' radical conceptuality came with the principle of virtual work–which provides what is considered the best a priori proof of the lever principle. First, it was assumed certain by the natural light of reason that the work to lift a body ten meters is just ten times the work to lift the body one meter. (And yet this is completely false by the inverse square law–right?) Work or energy is here made into a substance: which means among other things that it is conserved.

Proof by virtual displacement is an incoherent thought experiment in which the work done by the forces of a system as the system is displaced infinitesimally from equilibrium is evaluated and equated to zero. Infinitesimal calculus is required. The awkwardness of the technique is shown by the following quote from George Merrill, An Elementary Text-Book of Theoretical Mechanics, p. 121.

... there is no real motion or displacement, and no work is actually performed either upon the weight or by the applied force; but the result attained by the supposition that motion does take place, and that work is done, is virtually the same as if the machine really moved, in spite of the fact that it is in equilibrium.

The technique is taken seriously because it gives the same answers as the "regular" conditions for static equilibrium. In other words, the behavior of ideal phenomena is deduced from an impossibility, on the assumption that the phenomena's behavior remains lawful during the impossibility. This technique simply cannot be developed from common-sense or experiential premises. The technique is so incoherent that I won't try to connect it to the emerging picture of an actuality supported and guided by phantoms and phantom histories.

In passing, we see how conventionalistic and artificial physics' attribution of substantiality is. Virtual work makes energy into a substance. The law of inertia makes motion into a substance.

J. As I said in (F), I don't interpret possibility through probability because the latter only opens up yet another dimension of fantasy. But for completeness, I may review some of the early objections to probability theory. It was said that the toss of a coin should not have an outcome split equally between heads and tails, but rather a unique outcome determined by the laws of mechanics. The likelihood of the actual outcome is one and the likelihood of an unfulfilled prediction is zero.

We do not know before the fact whether the sun will rise tomorrow. There are two possible outcomes. If we argue solely from combinatorics, then the probability that the sun will rise tomorrow is 1/2. Moreover, there is a principle that a favorable run does not affect the next trial. On that basis, the favorable run of sunrises cannot increase the probability of tomorrow's sunrise above 1/2.

How do you get beyond the obvious observation that the probability of an actual occurrence is one and the probability of an unfulfilled prediction is zero? How can you ever assign a fractional likelihood to an event? Well, for throwing a die, the probability of a given number is one-sixth, because it is one of six possible outcomes. But an implicit quantification of probability has already been made here. The probability that the outcome will be "one" or ... or "six" is one; and each of the numbers is equiprobable. Well, yes, because the six faces of the die are geometrically interchangeable. The rough geometric symmetry of a coin or die guarantees that an outcome cannot be favored over its complements.


Section III

Taking their cue from Hume, various thinkers have attempted a complete demystification of physics: they propound rational reconstructions of physics which take sense-data as the starting point. Thus, to one degree or another, physics is pictured as a simple collation of sense-data. So we have, perhaps, Mach, Carnap's Logical Structure of the World, Albert Einstein's article for the Encyclopaedia Britannica, Jerome Rothstein in the September 1962 Physics Today. As we have seen in this manuscript, this enterprise was hopeless. Physics needs autonomously subsisting divine elements and events which can be produced (instantiated) at any time and place (subject to restrictions of regularities of Nature). Physics needs actual events to be wrapped in unrealized possible events. Physics needs metastability. Physics needs to start from the All, as Mach calls it. And then there is the matter of systematics and rules in-the-world. One does not obtain any of this by cumulating sense-data, or even by cumulating successive actual events (selections from history).

Empiricism is too weak to yield physics. Nevertheless, empiricism has produced an array of redefinitions which bring into prominence a zone of world-apprehension which was previously unexamined. (Indeed, it is quite common for scientific contributions to take this form.) If we explore the zone which the empiricists uncovered, without a prior loyalty to saving physics, we may deepen our escape from scientific reality, our annulment of physics at its base in common sense.

A. It is now sense-data which are supposed to be autonomously real; the regularities in the sensory realm comprise nature's order. Consider a highly counter-intuitive thought-experiment: let the regularities in the realm of sense-experiences be different from those our culture recognizes. How would this new situation be transferred upward into the construction of physics? Indeed, how would it be transferred upward into the construction of common sense? What is being attacked here is science's dependence on object-gestalts or hypostatized perceptions.

The Flynt lever prints a collinear Mueller-Lyer illusion on a lever. On the back is the same figure with a superimposed ruler.

Now the unloaded, uniform strip does not balance at its visual center. And when the ruler is added, the illusion dominates the scale. So the passive judgment of length becomes incoherent within a single mode, vision. Moreover, the concurrence of equilibrium and symmetry is disrupted phenomenally.

[Einstein in the Encyclopaedia Britannica: The notion of material body is more elementary than the notion of space. Material body: a conceptualization of the persistence in time, or continuity, of certain groups of experience-complexes. That the objects in perception seem to be material bodies is an illusion. Then space is the contact of material bodies. In perception, the touching of two bodies is inferred. (Einstein's text is reproduced as an appendix.)]

What if every time you saw one object, you felt two objects? Or what if every time you felt one object, you saw two? What would this do to the empiricist construction of reality? The given concept of material contiguity would be nullified. Two famous tenets would lose their naturalness: two point masses cannot be in the same place at the same time; one point mass cannot be in two separate places at the same time.

One point of this speculation. The empiricists want to take sense-experiences as irreducibly real. One then looks for regularities in the sense-data, and by cumulating them, one arrives at physics. If this is so, then empiricists should be equally prepared for an experience-realm which systematizes differently from the one we now acknowledge. Confronted with a different experience-realm, empiricists ought to be able to cumulate a new, unphysical world-pattern without a hitch. But I submit that Carnap would be lost in an experience-realm which failed to buttress his college mathematics, his notion of how time is objectified, etc.

The remainder of this section consists of notes to myself which tie the present discussion to other manuscripts. My finished exposition resumes in Section IV.

[Again Einstein: If time refers back to the subject's experience of ''now" and "before," then it is incoherent to ascribe time to an external object or process–to a weathered cliff-face or to the sun's daily transit.]

Suppose that I lived through episodes of my life more than once. Would time be circling? Am I conscious of the repeat as being in the future of the original episode? In dreaming, have I repeated episodes–and how literally is one episode a copy of the other?

A creature who had unbidden mental rehearsals of episodes before they occurred. A mirror-phenomenon to memory. How would this change the world-picture?

A shadow which is made to move by switching the light which casts it. Proceeding directly from observations of this shadow, there would be no motivation to posit the material point which retains its identity in motion. There wouldn't be any conservation of substance, except in the sense of the net sum of annihilation and creation. (Well, there were atomistic theories in the middle ages, but they referred to phenomena which apparitionally were continuously sustained.)

Construction, not of physics, but of common sense from sense- contents. Well, it's not just a matter of adding sense-data to make a pattern. The issue in perception of discerning the substantial configuration, in a perceptual field with aspects of a mirage. Being misled by reflective plate glass, for example. In fact, the "mirage" will contain tiny cues which establish the substantial configuration that it is veracious to infer. But if you have not been taught that these cues crucially disclose the substantial nature of the apparition, then you may see them without noticing them. Moreover, even noticing the cues is not enough. You have to be pre-indoctrinated in the theory of the substantial configuration before you can imaginatively synthesize it from the cues. There is a lengthy discussion in "Studies in the Person-World," §C. To see rightly when there are reflections, shadows, etc. means to inject a fantasy into the perceptual field which is derived from highly indirect considerations and conceptualization.

The lesson is that perception is a function of complete prior indoctrination. It infers from cues to the whole in utterly unsupported ways.

Suppose different people associate mutually exclusive event-series with the same time-period as defined by some external process such as one day equals sunrise to sunset. (Collating different people's dreams.)

Cases in which there is a different world for each subject–or in which the subject cannot be abstracted out of the world.

The notion of two material bodies in contact. Visually, one body begins where the other ends. Muscular manipulation of the bodies might disclose that the bodies cannot be crushed closer together. (But does this confuse separation with elasticity?) And yet visual evidence of abutment is worthless if one body can then slide in front of the other. What of a realm in which every time two bodies abutted visually, one could not slide in front of the other? (Well, that's like facing shapes in a plane.) If bodies could not pass through each other?

Again the Flynt lever (with superimposed ruler). Undermines the concept of distance because the illusion dominates the scale.

The waterfall illusion–negative afterimages of motion. The visual apparition of motion and the visual apparition of remaining in place are not mutually exclusive. Perception isn't confined to the logically possible.

If the common-sense notion of a substantial body is an intuition required to guide use of the notion of a point mass, what of the paradoxes of common sense concerning substantial bodies? Are the common-sense paradoxes transferred, hidden, to the physical notions? See "Paradoxes of Common Sense" (1988), §I.C.

In fact, physics is shown up as absurd by the Paradox of History from "Paradoxes of Common Sense." (typescript, page 6)

The possibility of a thing which is located in two mutually exclusive regions.

Does physics depend on a unity of knowing and willing? [Schopenhauer: Willing is not the material cause of action; willing and the material cause are two different perceptions of a single phenomenon. Willing implies a duality of "he who wills" and "what is willed"?]

My manuscript "An Epistemic Calculus." Reflections of optical illusions; etc. Empiricist collation of a physical order in the situations which I delimit in that manuscript. Beyond the first degree of illusion can there be a second degree of illusion?

Suppose that the conceptual vernacular, the first hypostatization, instead of positing "size constancy," said that semblances of things are real. E.g. things really get bigger as they move toward me, and really get smaller as they move away.

What we call a stationary object moves every time my visual vantage-point (my viewpoint) shifts.

Let us return to the problem of forming a class of recurrences of a thing (such as sunrises). Let the class include similar objects to perception, even if they don't have the same substantial basis. Then a table encountered in waking experience is classed with a table encountered in a dream. Or a relative who appears to you shortly after dying is classed with the living person. (To invoke the most common hallucination known to psychiatry.) Treating certain shadows as equivalent to free-standing objects.

The notion of positive whole numbers depends on having a succession of distinctive (distinct) events in time; or on material objects. Kneebone: discrete and identity-maintaining entities are required for positive whole numbers to be credible. Dependence of pure mathematics on the world-pattern. And yet: who says that raw experience provides discrete and identity-maintaining entities?

Again Jerome Rothstein on the objectification of time. The betweenness relation for remembered events: reduces to cumulation of events in memory. But: an interpolation, relating to an event which intrudes between two events you voluntarily attend to. Rothstein wants to derive the need for fractions in the arithmetic of the world from such interpolations.

On the contrary: fractions are abstractions which synthesize diverse concrete characteristics. Fractions have to apply equally well to time and to space and to material substances, masses.

Rothstein makes a deep distinction between events which you attend to voluntarily and events which intrude upon your consciousness (in characterizing how you experience the passage of time). But this picture equally could be a chronography of pure mentation interrupted by obsessive thoughts.

Versus voluntary punctuation of the time-stream by going away from place A and returning to it. A place literally can recur; a time cannot.

Free choice is tied to acting and to a space of more than one dimension. I can go from A and return to A through B, or not. But since time is one-directional and one-dimensional, you don't return to A, you advance to A'. Rothstein has not really proved that you could have acted differently at the time that you acted, that your life has a degree of freedom that the motion of a planet lacks. Physics must have free choice as an idea, but nothing "physical" proves free choice.

Extrapolation of time to before my birth and after my death.

Rothstein's account of the history of science provides for no conceptual dislocation concomitant with a clash of societies or cultures. Realistically, the ancient Mediterranean public and social situation had to be transformed ideologically before the concept of mass could arise, and before there could be an explosion in the scientific use of the infinty concept [before natural philosophers would give a rationale for the infinite and deduce the world from the infinite].

As I said earlier, those who "construct physics from experience" are in fact imposing their physics indoctrination on selected pre-integrated "lived experience" such as a steam engine. They have forgotten the structure of "intuitive experience"–or, they militantly negate intuitive experience, as when they posit frictionless processes.

Let us look not at physics but at the intuitive apprehension of time. For one thing, we presume that day/night alternation–or an individual's pulse-rate–is relatively isochronous over a lifetime. On the other hand, duration-judgments have a "psychophysical" feature. "One month" ten years ago is subjectively shorter than "last month"–perhaps by an order of magnitude. (This does not persist uniformly to the present moment.)

So the subjective temporal metric is obdurately inconsistent. We live in experiential inconsistency.

Let me observe the minute hand of an electric motor clock, while feeling my pulse to demarcate time-intervals. I start observing when the tip of the minute hand is at some chosen mark on the clock face. Let the mark's position be m. Let the minute hand's position at successive pulse-beats 1, 2, 3, ... be h1, h2, h3, ... Let = mean "not discriminably different." I find these successive relationships:

h1 = m

h2 = m h2 = h1

h3 = m h3 = h2

h4 = m h4 = h3

h5 m h5 = h4

We have perception of two extra-mental series of "events." We find the logical principle that sameness is transitive to be violated in the case of slow change. Here again we have a feature of experience which Carnap et al. don't want in physics–least of all in mechanics.

B. It could be thought that the foregoing speculations are pointless, since the experience-realm does not manifest the anomalous patterns discussed. But in fact every anomaly I have hypothecated is provided in a perceptual illusion or other supposedly deceptive episode in lived experience. Now: what if the experiences were accepted as elementarily real, without the corrective of knowledge of the substantial configurations? What world-pattern would they cumulate to?

Indeed. Physics cannot be inferred from anything tangible or anything tautologous. There must be a cosmic fantasy which is absolutely transcendental and which is accepted on absolutely blind faith: in order to warp the bits of experience into the abstract laws.

And there is no reason to suppose that the prevailing cosmic fantasies are unavoidable. In a future study, I will examine closely the rational progression from classical mechanics to classical pure thermodynamics. This progression is altogether jerry-rigged. The story called physics was written not by a god but by an opportunist.

As for my aim of annulling physical reality, these remarks accomplish that aim in principle. To pass to the operative consequences of the cancellation of physical reality, however, is a project for the centuries; and this manuscript can make only a beginning in that direction.


Section IV

Let me now consider some elementary successes of the physical method-as I promised in I.C.

A. The proof that the diagonal of a square with side = 1 is 2. Counting congruent triangles on the "Diamond Diagram." (Cf. my "Was Greek Mathematics Crazy?'' typescript, p. 9.)

This is a geometric question, entirely separate from the question of whether 2 can be expressed as a quotient of whole numbers. By the Euclidian method, you have to follow the diagrammatic proof (appreciate the proof) perceptually. (Modern mathematics tells you how to abstract a numerical lattice from a geometrical figure, but it doesn't tell you how to recover the perceived array from the numerical lattice.)

Dijksterhuis speaks of "as extreme an idealization as that by means of which Euclidian geometry is deduced from physical experiences of solid bodies." Well, well! So Euclidian geometry's degree of reality is identical to that of Newtonian mechanics. In any case, let me focus on the operational problem. To realize the proof, you have to be able to establish objective position operationally. This means drawing: which is performed with rigid equipment and guided by sight and touch.

An optical illusion not so different from the diamond diagram makes the point that judgment of length by sight alone is unacceptable.

B. The identically symmetrical lever: unloaded arms or arms with equal weights at the two ends. A vindication of sufficient reason and symmetry. (Remember that at a more advanced level of physics, symmetry seems to be violated by the deflection of a magnetized needle suspended parallel to a wire through which an electric current is passed.)

Note the "phenomenal" zone. The lever is assumed immobile even though it moves in my visual field. The fulcrum is assumed to substantially bear the lever, even though sight only provides that the lever is atop the fulcrum.

Answer experiments. The Flynt lever. The shadow lever. A hung bar casts a lever-like shadow on a fulcrum. (At last a lever which is infinitely thin and weightless.)

C. A unison monochord. Equal lengths of a uniform string sound the same pitch. Sufficient reason; symmetry.

Here is a case where refined perception can be applied. Do the pitches defined by bisecting the string geometrically really sound a unison?

Does the person know where the unison is, even if not an expert singer?

D. Rolling a ball against a backstop. The angles of incidence and rebound are equal. The most elementary demonstration of least action.

A single, primitive trial is relied on to exemplify a vast and significant family of phenomena. At the same time, an actual trial is likely to find the angles perceptibly unequal. Physics can improve the result by leveling the floor, but that does not prove that the outcome is not influenced microscopically by phenomena which have not yet been mentioned.

E. Establishment of isochronic intervals and the isochrony of the pendulum. Start a pendulum swinging and synchronize a metronome with its swings. They remain synchronized until the pendulum stops. So both the constancy of the pendulum's period and the equality of the simultaneously demarcated time-intervals are validated.

(By relativity, the pendulum and metronome are not absolutely synchronized. Advanced physics undermines the truths of inspection or operational truths by which physical notions are given meaning in the first place.)

Ingenuous thought does not understand the metronome's mechanism. But all that matters here is that the metronome's operation not cause or be caused by the pendulum's operation–and that is plausible.

A single, primitive trial is relied on to exemplify a vast and significant family of phenomena. Or to put it another way, if an actual pendulum were now found which was not isochronic, it would be blamed on its failure to be a perfect pendulum.

"Phenomenology." Could the time-unit itself expand and contract, and do so at different rates for different processes, within the perspective of a single observer? Here is where the principle of cognitive parsimony is wanted, to rebuff such speculation.

Answer experiment. The shadow pendulum. The shadow on a wall of a pendulum. Since the period of a pendulum is independent of the weight of the bob in any case, the shadow pendulum establishes the law of the pendulum as well as the substantial pendulum. Proportionality of period to square root of length is borne out.

In other words, the shadow pendulum conforms to the mechanical law of the pendulum, which is indifferent to the pendulum's weight and which does not ask whether the pendulum is a substantial body. An empiricism which relies on vision will find the shadow pendulum to obey the mechanical law of the pendulum. Regularity does not assure substantiality. (Cf. II.E.) The knowledge that the shadow is an insubstantial satellite of a substantial body pinned in a light beam comes from common sense; it is a fantasy-laden integration of extra-mental reality.

F. A ball, moving freely from the same distance above a plane, reaches the same instantaneous speed as it hits the plane-whether it falls or whether it rolls down a gentle incline.

This equality of terminal speeds was originally considered to be a paradox, since the travel times are different.

The meaning of instantaneous speed. In the first place, maximum speed is assigned to the falling ball at the instant it comes to rest (or rebounds).

Before infinitesimals, instantaneous speed was defined as the speed an object would have if it continued uniformly from the point in question. So a vicious circle. Subsequently instantaneous speed comes to have the form 0 ÷ 0. The experiment proves 0 ÷ 0 = 0 ÷ 0.

To guarantee definite solutions of 0 ÷ 0, must the continuum of real numbers be actual? Building an inclined plane so the ball rolls off at 2 m/sec. (This result means nothing because it depends on empirical g?)

G. A ball rolling on a V-plane travels to the height from which it started.

When the ball rolls uphill, it does "work." So the experiment graphically illustrates work, the notion which mechanics bequeaths to thermodynamics. The experiment is cited to motivate or prove the notions of reversibility and conservation of mechanical energy–even though it precisely belies them.

H. The lever with weights in small whole-number ratios. The weight-ratio is validated by using geometry to measure different volumes of the same material. To conduct the experiment, one needs the lightest, thinnest lever which remains rigid. Place the lighter weight at the end of one arm. Then place the heavier weight on the other arm where a balance is produced. Demonstrates the law of the static moment,

W1L1 = W2L2

Thus equilibrium is shown to be a state of symmetry. Note that attempts to prove the equation directly from sufficient reason are considered specious. But that then means that the relationship is only known empirically, not deductively.

I. Irrational ratios in the real world. Lever with arms and weights in ratios of 1:2. The arm ratios and the weight ratios can be established separately via geometry. But what should be noted is that W1L1 = W2L2 is only known empirically. Its deduction from first principles is a hoax. Thus, the formula has not been established deductively to hold for irrational values.

Analog multiplication of irrational quantities in different physical modes to produce a perfect integer.

J. The tritone monochord. The sensuous phenomenon called a musical interval now supports or exemplifies an irrational ratio. Something ancient Greek thought did not wish to acknowledge. The composite wave form takes forever to cycle out-but this sort of analysis is very modern.

K. Physical experiences with infinity. The composite wave form of a tritone. Galileo: A body falling from rest passes through all the infinite degrees of slowness before attaining any positive speed.


Section V

A. I now wish to examine one of the great canards spawned by early twentieth century science: the tenet that there can be mathematical theories which are true and yet inoperative in the real world. This canard evidently stems from i) Hilbert's rationale for geometry; ii) Einstein's thesis that the geometry of the objective world is elliptical, not Euclidian; iii) Hilbert's formalist program for validating arithmetic (mathematical induction). Concurrently, the notion spread that geometry was not about any actual space but about artificial spaces determined by axiom systems.

According to the canard, a mathematical system is an uninterpreted calculus–the way chess could be said to be an uninterpreted calculus. Mathematical "truth" consists in nothing more nor less than the nonderivability of a formal contradiction (A&¬A) in the system: that is, consistency. Hilbert: "consistency is existence." Consistency may be proved by interpreting the formal calculus so that its propositions become propositions of some branch of mathematics whose consistency is accepted on trust.

Hilbert's scheme received one professionally accepted defeat. The profession accepted the Incompleteness Theorems–the absolute consistency of arithmetic could not be proved as Hilbert had wanted.

There were other challenges from the fringes. Brouwer claimed an inconsistency proof for classical analysis. But the profession never accepted that his exercise was an inconsistency proof; and Brouwer's intuitionism was assimilated to classical mathematics (as constructive mathematics). Wittgenstein was another eccentric. (Wittgenstein said, by the way, that intuitionism was all bosh.) Wittgenstein argued vehemently against Cantor's higher degrees of infinity; but the profession ignored Wittgenstein's objections.

The Incompleteness Theorems have a cultural import, against formalism, which should not be underestimated. The rudiments of an axiomatic "game" can be recursively defined. Nevertheless, if the game is infinitary, then its realizations may involve mysteries which can never be penetrated. The upshot is the twentieth-century metamathematical picture of mathematics as mechanical at the bottom, mystical at the top. And yet, the practice of codifying mathematics as an unfolding of arbitrary axiom-systems has become ubiquitous. In that sense, Hilbert's epistemology still provides the style of modern mathematics–it is just that the consistency of elementary branches is supposed to be taken on trust.

B. What I propose to do here is to examine these canards in conjunction with the preceding discussion of i) the physical world as an idealization, and ii) the phenomenological and operational prerequisites for physics. This is a perspective on the topic opposite to that of metamathematics and foundations of physics. As Wittgenstein indicated when positioning his remarks on mathematics, professional science regards perceptual and operational matters as something to be taken care of in childhood. It regards it as childish to recur to them in speculative research. Be that as it may, I am committed to a supersession of modern science which requires that phenomenological and operational matters be rethought.

C. What we have seen, in the first place, is that physics' real world and intuitive pure geometry (Euclidian geometry) are already "idealizations."

The theory of motion calls for just as extreme an idealization as that by means of which Euclidian geometry is deduced from physical experiences of solid bodies.


Dijksterhuis doesn't know the half of it. There is no question of deducing Euclidian geometry from physical experiences of solid bodies: the basics of the physical world were not codified until Newton–and classical mechanics is if anything more artificial than intuitive pure geometry. The reader may review my commentary on the principle of virtual work, II.I. Euclidian geometry is not "deduced" from any more elemental realm of experience. What Euclid invented, or codified, was a new way of being insane: a way of integrating vision, touch, rigid or inelastic object-gestalts, and cognition of extra-empirical exactness via logic. As one learns the system and uses it, one learns to compensate unconsciously for the discrepancy between material isometry and the foreshortening which characterizes vision (an indispensable faculty in following Euclidian proofs). (Again, cf. the diamond and corner diagrams; and cf. the notorious curving of the Parthenon to make it look straight.) Geometry is perceptually false. Refined perception apprehends that the tangent to an arc never touches the arc in only an unextended point. Geometry is conceptually implausible. One has to believe that two circles of radius r, with centers separated by 2 r, intersect in angles which become right angles just at the unextended point of intersection. When a cylinder rolls off its own area on a plane, one has to believe that parallel line segments are somehow placed contiguously and cumulate to form a positive area–even though they individually have zero width. (Yes, I know the mainstream twentieth-century resolution of this difficulty; the point is that for over two thousand years mathematicians believed without a rationale.) And we are supposed to believe that a curve can approach ever more closely to its asymptote without ever reaching it–a fantasy which, by the way, begs the question whether the universe is infinite.

Turning to classical physics, we may ask the most obvious question: what is the geometry and physics of the collision of two point masses?

The best mathematics and the best physics are incoherent multi-modal syntheses or melanges. Dijksterhuis has adequately noted the absurdity of the point mass moving in infinite empty space. Then, we must remember that classical mechanics begins by dismissing all dissipative and dispersive phenomena: thus negating the most obvious generalizations about lived experience.

D. We began with the topic of the mathematical theory which is true and yet inoperative in the real world. Well, speaking universally, this dichotomy of the theoretical and the real turns out to be ironic and vacuous. All of the worlds which science provides are imaginary (and incoherent). But that does not mean that the question about mathematical truth and the real world has nothing to teach us. The question admits of restricted or relative answers which may in fact contribute to my enterprise of superseding inherited physico-mathematical science.

1. Hilbert and metamathematics to the contrary, there is no mathematical theory which is a "game" in the sense that chess is–for a reason which has never before been noted. That is that to have a bona fide game, it is necessary to codify the rules in natural language. Mathematicians will not (and cannot) be forthcoming in this way. Natural language cannot be the bottom of mathematics.

2. A study of the vicissitudes of inconsistency relative to fashionable mathematical doctrines shows that consistency does not serve as the guiding goal of mathematical thought–no matter how much has been made of it since Hilbert said "consistency is existence." Somehow mathematical ideas become popular independently of consistency. Consistency is then endlessly redefined or finessed after the fact to deflect certain objections to a popular idea.

3. Even though Hilbert and others wanted Euclidian and non-Euclidian geometry to have the same status in pure mathematics–and even though Euclidian geometry already violates sensuous-concrete experience–there is an important respect in which non-Euclidian geometry is more artificial than Euclidian geometry. (And this even though of course physics regards non-Euclidian geometry as more realistic.) Euclidian geometry has a background of intuitions (tortuous though they may be)–and of manipulations which are required to follow diagrammatic proofs perceptually–which one painfully learns upon one's first exposure to mathematics. As for elliptical geometry, both its explanations and its consistency proof reinterpret its propositions as propositions of e.g. spherical trigonometry. Elliptical geometry has no independent intuitive-operational background. It is grafted onto the Euclidian background; in that sense it is parasitic on Euclidian geometry. (This remains so even though intrinsic properties can be defined which would reveal if we were physically confined to a curved manifold. The key step, by the way, is to establish the physical definition of a geodesic.) Given the Euclidian notion of a plane, and of parallel lines extrapolated indefinitely, no intuitive picture has even been provided of how the parallel lines could meet. (Another way of looking at it is that the cogency of elliptical geometry only shows that the traditional axioms were an inadequate means to specify traditional geometry.)

The reason why my observation here is not a commonplace is that modern science cannot acknowledge Euclidian geometry's intuitive and operational background. Again, modern science insists that we have left the humble phase of subjective appreciation and comprehension behind.

4. Archimedes said that a lever can bear weights in an irrational ratio. Galileo said that a body falling from rest passes through all the infinite degrees of slowness before attaining any positive speed. So we have certainly had claims that the irrational and the infinitely divisible are actualized in the physical world.

In any case, it is said, real analysis is a cogent theory, whether it is literally faithful to the physical world or whether the assumptions of infinite divisibility and continuity are only a computational convenience.

But now there is a possibility of an attack on mathematics from an unexpected direction. The addition of the technology of amplification and of oscillators to the already existing string and keyboard musical instruments makes it possible to cultivate the perceptual judgment of pitch intervals in a thorough way. To a completely new degree, measurement can become a perceptual enterprise–called tuning. The lessons of perceptual judgment can then supply a serious reason to reject certain abstract speculations as meaningless.

To repeat, various developments have deepened the enterprise of tuning. Oscillators supply pure and stable tones. Amplification makes difference tones and beats obtrusively discernible. Before that, the piano provided a machine which was supposed to be preset to irrational intervals. This then led to a certain understanding of what is involved in finding an irrational interval.

Somebody who works with these modern means, and pursues the enterprise of tuning, may conclude that irrational intervals cannot in fact be reached by the method of judgment called tuning. If that could be a defensible position, then it would be possible to transfer that lesson to the realm of lengths, and to transfer it back into the idealization called Euclidian geometry. The computation that a square of side equal to one has a diagonal of length 2 would then be rebutted. This could become the first redirection of pure mathematics on cultural grounds.

On the other hand, I have to note that the recent cultivation of tuning is a fetishism in the sense that the musical use of pitch need not follow scale theory (or physics) exactly. The soloist in a violin concerto plays octaves sharp to make them noticeable to the audience. Classical violinists play leading tones sharper than the definition of the scale provides for. The intonation of an orchestra playing Wagner is presumably guided by incompatible considerations.

Moreover, I am not yet convinced that the tritone is not capable of sensuous-qualitative recognition–after all, it exactly splits the octave. Difficulty in finding the tritone perceptually could just be a self-imposed impairment.

• • •


R. H. L. Avenarius, Kritik der reinen Ehfahrung (1888)

A. J. Ayer in Problems in the Philosophy of Science, ed. Imre Lakatos and Alan Musgrave (1968), p. 164 ("Are all our common sense judgments false? ... I have great sympathy and admiration for anyone who will come right out and say that the common-sense view of the world is just untrue ...")

David Bloor, Knowledge and Social Imagery (1976), especially

Ch. 6: Can there be an alternative mathematics?

Ch. 7: Negotiation in logical and mathematical thought

Ludwig Boltzmann, Theoretical Physics and Philosophical Problems (tr. 1974), especially page 231

Leon Brillouin, Science and Information Theory, 2nd ed. (1962)

L. E. J. Brouwer, "Mathematik, Wissenschaft und Sprache" (1929), in L. E. J. Brouwer, Collected Works, Vol. 1, ed. A. Heyting (1975)

Buck & Macaulay, Maximum Entropy in Action (1991)

Wendell Bush, Avenarius and the Standpoint of Pure Experience

Rudolf Carnap, The Logical Structure of the World (tr. 1967)

Rudolf Carnap, Logical Foundations of Probability (1950)

Ernst Cassirer, The Philosophy of Symbolic Forms, especially Vol. 1, pp. 75-6, 85, 108-9; Vol. 3, Part III, Ch. 5

Paul Chambadal, Paradoxes of Physics (1971; tr. 1973)

Noam Chomsky, Reflections on Language (1975)

K.G. Denbigh and J.S. Denbigh, Entropy in Relation to Incomplete Knowledge (1985)

Martin Deutsch, "Evidence and Inference in Nuclear Research," in Evidence and Inference, ed. Daniel Lerner (1959), pp. 96-106

E. J. Dijksterhuis, The Mechanization of the World Picture (tr. 1961)

Michael Dummett, Truth and Other Enigmas (Cambridge, 1978), page 268 (there are no phenomenal qualities and all observational predicates are inconsistent)

Albert Einstein, "Relativity Theory," in "Physical Theories, Mathematical Aspects of," Encyclopaedia Britannica, Macropaedia Vol. 14 (15th edition, 1984) ("All thoughts and concepts are called up by sense experiences and have a meaning only in reference to those sense experiences."–etc.)

P. and T. Ehrenfest, The Conceptual Foundations of the Statistical Approach in Mechanics (1912; tr. 1959)

Galileo Galilei, Two New Sciences (tr. Stillman Drake)

Galileo Galilei, Dialogue on Two World Systems (tr. Stillman Drake)

T. Gold, "The Arrow of Time," American Journal of Physics, 1962, p. 403

Nelson Goodman, The Structure of Appearance (1951)

T. L. Hankins, Jean D'Alembert: Science and the Enlightenment (1970), Chs. 7-10

Alan Hausman & Fred Wilson, Carnap and Goodman: Two Formalists (1967)

Heinrich Hertz, The Principles of Mechanics (Dover, 1956), especially "Introduction"

Edmund Husserl, The Phenomenology of Internal Time-Consciousness (tr. 1964)

Edmund Husserl, The Crisis of European Sciences and Transcendental Phenomenology (tr. 1970)

Norman Kretzmann, ed., Infinity and Continuity in Ancient and Medieval Thought (Cornell, 1982)

Moses Maimonides, Guide for the Perplexed (Dover, 1956), page 120

Ernst Mach, Contributions to the Analysis of the Sensations (tr. 1984)

Ernst Mach, The Science of Mechanics (2nd English edition, 1919), pp. 13-14 (compare 5th English edition, 1942, pp. 19-27)

Ernst Mach, Erkenntnis und Irrtum (1906)

Ernst Mach, History and Root of the Principle of the Conservation Of Energy (tr. 1910)

Ernst Mach, "Uber einige Hauptfragen der Physik" (1868)

Ernst Mach, Physicist and Philosopher, ed. Robert S. Cohen and Raymond J. Seeger (1970)

Robert S. Cohen, "Ernst Mach: Physics, Perception and the Philosophy of Science," Synthese 18, p. 132

A. Koslow, "Mach's Concept of Mass: Program and Definition," Synthese 18, p. 216

Maurice Merleau-Ponty, Phenomenology of Perception (tr. 1962)

George Merrill, An Elementary Text-Book of Theoretical Mechanics (1905)

Karl Popper, Quantum Theory and the Schism in Physics (1982)

Emil Post, "Absolutely Unsolvable Problems and Relatively Undecidable Propositions–Account of an Anticipation," in The Undecidable, ed. Martin Davis (1965)

Frank P. Ramsey, The Foundations of Mathematics (1931), pp. 20-21 (mathematics has nothing to do with thought, language, or symbolism)

Jerome R. Ravetz, Scientific Knowledge and Its Social Problems (1971), especially page 218

Jerome Rothstein, in Physics Today (Sept. 1962), pp. 28-38

Jerome Rothstein, "Informational Generalization of Entropy in Physics," Quantum Theory and Beyond, ed. Ted Bastin (1971)

H. A. Simon et al., "Scientific Discovery as Problem Solving," Synthese 47 (1981), pp. 23-5

Lawrence Sklar, Physics and Chance (1993), Ch. 11

Hermann Weyl, 5pace-Time-Matter (Dover), especially "Introduction" (1950)

• • •


All thoughts and concepts are called up by sense experiences and have a meaning only in reference to these sense experiences. On the other hand, however, they are products of the spontaneous activity of the human mind; they are thus in no wise logical consequences of the contents of these sense experiences. Therefore, to grasp the essence of a complex of abstract notions it is necessary for the one part to investigate the mutual relationships between the concepts and the assertions made about them; for the other, to investigate how they are related to the experiences.

As far as the way is concerned in which concepts are connected with one another and with the experiences, there is no difference of principle between the concept systems of science and those of daily life. The concept systems of science have grown out of those of daily life and have been modified and completed according to the objects and purposes of the science in question.

The more universal a concept is, the more frequently it enters into human thought; and the more indirect its relation to sense experience, the more difficult it is to comprehend its meaning; this is particularly the case with prescientific concepts that are customarily used from the time of childhood. Consideration of the concepts referred to in the words "where," "when," "why," "being," to the elucidation of which innumerable volumes of philosophy nave been devoted, illustrate the difficulty. The philosopher fares no better in speculations than a fish which should strive to become clear as to what is water.

Time. The physical time concept derives from the time concept of the nonscientific mind. Now, the latter has its root in the time order of the experiences of the individual, and this order must be accepted as something primarily given. Man experiences the moment "now," or, expressed more accurately, the present sense experience combined with the recollection of (earlier) sense experiences. That is why the sense experiences seem to form a series, namely the time series indicated by "earlier" and "later." The experience series is thought of as a one-dimensional continuum. Experience series can repeat themselves and can then be recognized. They can also be repeated inexactly, wherein some events are replaced by others without the character of the repetition becoming lost. In this way man forms the time concept as a one-dimensional frame that can be filled in by experiences in various ways. The same series of experiences answer to the same subjective time intervals.

The transition from this "subjective" time to the time-concept of prescientific thought is connected with the formation of the idea that there is a real external world independent of the subject. In this sense the (objective) event is made to correspond with the subjective experience. In the same sense, there is attributed to the "subjective" time of the experience a "time" of the corresponding "objective" event. In contrast with experiences, external events and their order in time claim validity for all subjects.

This process of objectification would encounter no difficulties were the time order of the experiences corresponding to a series of external events the same for all individuals. In the case of the immediate visual perceptions of man's daily life, this correspondence is exact. That is why the idea that there is an objective time order became established to an extraordinary extent. In working out the idea of an objective world of external events in greater detail, it was found necessary to make events and experiences depend on each other in a more complicated way. This was at first done by means of rules and modes of thought instinctively gained, in which the conception of space plays a particularly prominent part. This process of refinement leads ultimately to natural science.

The measurement of time is effected by means of clocks. A clock is a thing which automatically passes in succession through a (practically) equal series of events (period). The number of periods (clock time) elapsed serves as a measure of time. The meaning of this definition is at once clear if the event occurs in the immediate vicinity of the clock in space; for all observers then observe the same clock time simultaneously with the event (by means of the eye) independently of their position. It was assumed that the conception of simultaneity had an absolute objective meaning also for events separated in space.

Space. In the present section the meaning of "where," that is, of space, is of concern. It appears that there is no quality contained in individual primitive sense experiences that may be designated as spatial. Rather, what is spatial appears to be a sort of order of the material objects of experience. The concept "material object" must therefore be available if concepts concerning space are to be possible. It is the logically primary concept. This is easily seen if an analysis is made of the spatial concepts, for example, "next to," "touch," and so forth, that is, if their equivalents in experience are sought. The concept "object" is a means of taking into account the persistence in time or the continuity, respectively, of certain groups of experience complexes. The existence of objects is thus of a conceptual nature, and the meaning of the concepts of objects depends wholly on their being connected (intuitively) with groups of elementary sense experiences. This connection is the basis of the illusion that makes primitive experience appear to provide information directly about the relation of material bodies (which exist, after all, only insofar as they are thought).

In the sense thus indicated there is (the indirect) experience of the contact of two bodies. Attention only is called to this; nothing is gained for the present purpose by singling out the individual experiences to which this assertion alludes. Many bodies can be brought into permanent contact with one another in manifold ways. In this sense the term "position relationships" of bodies is used. The general laws of such position relationships are essentially the concern of geometry. This holds, at least, if it is not desired to regard the propositions that occur in this branch of knowledge merely as relationships between empty words that have been set up according to certain principles.

Now, it is important to examine the meaning of the concept "space" which is also encountered in prescientific thought. The concept of space in prescientific thought is characterized by the sentence: "It is possible to think away things but not the space which they occupy." It is as if, without having had experience of any sort, man had a concept, even a presentation, of space; and as if he ordered his sense experiences with the help of this concept present [sic], a priori. On the other hand, space appears as a physical reality, as a thing which exists independently of thought, like material objects. Under the influence of this view of space the fundamental concepts of geometry, the point, the straight line, the plane, were even regarded as having a self-evident character. The fundamental principles that deal with these configurations were regarded as being necessarily valid and as having at the same time an objective content. No scruples were felt about ascribing an objective meaning to such statements as "three empirically given bodies (practically infinitely small) lie on one straight line," without demanding a physical definition for such an assertion. This blind faith in evidence and in the immediately real meaning of the concepts and propositions of geometry became uncertain only after non-Euclidian geometry had been introduced.

If a beginning is made from the view that all spatial concepts are related to contact experiences of solid bodies, it is easy to understand how the concept "space" originated, namely, how a thing independent of bodies and yet embodying their position possibilities was posited. If there is a system of bodies in contact and at rest relatively to one another, some can be replaced by others. This property of allowing substitution is interpreted as "available space." Space denotes the property in virtue of which rigid bodies can occupy different positions. The view that space is something with a unity of its own is perhaps due to the circumstance that in prescientific thought all positions of bodies were referred to one body (reference body), namely the earth. In scientific thought the earth is represented by the coordinate system. The assertion that it would be possible to place an unlimited number of bodies next to one another denotes that space is infinite. In prescientific thought the concepts "space" and "time" and "body of reference" are scarcely differentiated at all. A place or point in space is always taken to mean a material point on a body of reference.

If Euclidean geometry is considered, it is seen that it refers to the laws regulating the positions of rigid bodies. It turns to account the ingenious thought of tracing back all relations concerning bodies and their relative positions to the very simple concept "distance." Distance denotes a rigid body on which two material points (marks) have been specified. The concept of the equality of distances (and angles) refers to experiments involving coincidences; the same remarks apply to the theorems on congruence. Now, Euclidean geometry, in the form in which it has been handed down from Euclid, uses the fundamental concepts "straight line" and "plane" which do not appear to correspond, or at any rate, not so directly, with experiences concerning the position of rigid bodies. (On this it must be remarked that the concept of the straight line may be reduced to that of distance. A hint of this is contained in the theorem: "The straight line is the shortest connection between two points." This theorem served well as a definition of the straight line, although the definition played no part in the logical texture of the deductions.) Moreover, geometricians were less concerned with bringing out the relation of their fundamental concepts to experience than with deducing logically the geometrical propositions from a few axioms enunciated at the outset.

Albert Einstein, "Relativity Theory," in "Physical Theories, Mathematical Aspects of," Encyclopaedia Britannica, Macropaedia Vol. 14 (15th edition, 1984)