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COMMON SENSE ANALYZED AS A PARADOXICAL THEORY

1977/2007

Common sense is conceptualization “tuned” or “pitched” so that it serves as the thought-medium of everyday life and ordinary social interaction. Everybody is familiar with it; to codify it should produce a shock of recognition.

One of the chief features of common sense is the incorporation of objective and subjective in the same scheme, in their interdependence. So it is that we may call a person responsible for a material act; so it is that our labels for time are based on our experiential window on it (now, the past, the future).

The codification of common sense offered here is probably logically circular. That is not a fault as long as our concern is to spell it out and explore it. In fact, the exposition is so circular that paradoxes surface already in the definitions and postulates. That does not diminish the codification as an anthropology of vernacular thought.

Common sense does not compete with science; science presupposes common sense and overlays it with arcana without in any way annulling it. If you don’t know object-gestalts and don’t understand words, then: you cannot arrive at the laboratory or lecture-hall doorway and enter it, nor can you make an appointment at the laboratory or lecture hall, nor can you read book titles.

Scientists may claim solutions to the paradoxes of common sense (solving the continuum with nondenumerable infinity, not to say the first three Souslin conditions). They may claim to find realities which violate common sense (virtual particles). But the overwhelming majority of people never learn large cardinals or quantum field theory—even as they live everyday lives. (I submit that common sense does not contain the mathematical apparatus which distinguishes fractions from reals.) Moreover, science does not even pretend to have a substitute for the common sense of awareness and responsibility, for example. It “solves” these aspects of life by denying them outright.

Incidentally, the arcana of science do not themselves escape paradox, or dispute. Claims made for the benefit of the laity that the paradoxes have been resolved are outright lies trading on the laity’s ignorance of the advanced literature.

Husserl argued that the present is not instantaneous. That may be a stimulating observation, but it does not eliminate the paradxoes, because common sense demands that time have a geometry. (In other words, a mechanical clock is a commonplace; its procession is customarily circular but could be rectilinear).

Occult or magical belief may be widespread, but we do not incorporate it here because common sense is the conceptual medium which is practical. A magician may say that he can conjure a ship out of nothing by saying a word or waving a wand. But that is not reliable, and meanwhile ships are built by prolonged and demanding work. (Proportionate causation.) Common sense addresses that side of things. On the other hand, common sense must recognize the phenomena of awareness and responsibility. To what degree that is already occult, or beomes occult, is a matter of judgment. Life after death is not incorporated here even though many peoople believe in it. (Actually they don’t know what they believe; most people treat the Bible as their supernaturalism textbook but are remarkably ignorant about what is says.)

In this 2007 revision of the 1977 version, paradoxes in pure mathematics without urgent physical consequences are omitted even if common sense could comprehend them. (E.g. Galileo’s paraodox, Berry’s Paradox.) They will be found in “Paradoxes of Naive Mathematics.”

In this version, I anounce definitions and then postulates of common sense. A paradox of common sense may then be formulated as a conclusion from one (or more) postulates which contradicts another postulate. We write “¬ Postulate 14,” for example.

I wrote the present revision in order that Paradigm 1—also from 1977—could be updated. But it is only a stopgap. It needs to be combined, for example, with the 1988 version, which until now was the definitive version.

DEFINITIONS

General remark. Here we do not count a fantasized entity or occurrence as an entity or occurrence. That is why we avoid the adjective ‘real’.

General remark. Some definitions are deferred to the postulates, because they cannot be separated from dogmatic claims.

An entity is substance (in physics, “mass,” but that is tendentious) occupying a volume of space and persisting in time. (Which assumes passage of time.) It has an identity: which is determined by convention and causation. [The relation between object-identity and causation may well be circular.]

(Identity, being determined by convention and causation, is subject to complications: the identity of an ice cube which has shed one drop of water; the identity of a living thing which persists as it consumes from its surroundings and exudes to its surroundings.)

A point is an element of a figure (geometry), or of an entity, that is without extension.

A continuous extent is an extent without gaps. A continuous entity is one that has no points at the entity’s location not belonging to the entity. Obviously an entity can be continuous along one axis and not another. Common sense can analyze this notion in a limited way. It means that if two distinct points are elements of a line or continuous entity, then there is a point between them which is also an element. Limitless divisibility, then. Beyond that common sense does not go.

Motion of an entity presupposes passage of time. It is the occupation by the entity, continuously in time, of different locations continuous in space.

Change of an entity presupposes passage of time; the entity becomes different qualitatively. A person ages; a flower wilts. Again, whether the entity foregoes one identity for another identity is a matter of convention and causation. When we posit that the entity retains its identity, then one and the same entity is constituted by two mutually exclusive totalities (sometimes called phases). [This is already a paradox in the definitions.]

An event or occurrence is a motion or change which is demarcated conventionally in time and space. It is essentially defined by its time-interval. “Instrumental” occurrences can have copies; historical occurrences may be asserted not to have copies. (There was only one assassination of President Lincoln.)

The universe is an extrapolation of the notion of entity without preconceived limits. (1) The extrapolation of space and its contents as we apprehend them. Its temporal extent left undefined. Or: (2) The extrapolation of the whole continuum and its contents as we apprehend them. (Three spatial dimensions and a non-interacting time dimension. Now all of time is meant.)

My awareness comprises, for example, perception, visualization, silent verbalization (verbal thought), imagination (which may combine visualization and verbal thought), wish (imagination with commitment), will (a wish for my act which immediately ensues), the capacity to espouse propositions (belief). [Profoundly important constituents of awareness are not mentioned here because Western thought makes knowledge and objectivity correlative, and therefore does not consider major constituents of awareness relevant to knowledge. If we added postulates for emotion, value-judgments, mood, etc., we would leave the channel of Western thought.]

The present is the instant of time I perceive; time is relativized to my immediate knowledge.

A person’s awareness, anchored to a definite body, comprises a personal identity even though interrupted by periods of unconsciousness. A person’s awareness as life-long personal identity is called the person’s mind.

A (natural) language is an invented and learned medium for embodying knowledge in conventional tokens, thus making knowledge communicable.

A veridical proposition is a sentence which codifies knowledge. When a veridical proposition is subjected to the syntactical operation of negation, the result is a false proposition.

[The perspective implicit in common sense is that knowledge is something in awareness that is correlative to reality. But if you want to see that spelled out, it goes far beyond common sense. Analytic philosophy is worthless in this connection, and one has to turn to a philosophical tradition long out of fashion. If one gives the requested definition of knowledge, most educated people will ridicule it as “windy words.” Very well, but what is expressed by these windy words is the stuff of life of the people who ridicule them. It is far beyond the topic here, but educated people operate in a frame of reference about which they vehemently refuse to be forthcoming. Another example is provided by “the moral responsibility of the individual.”]

A hallucination is a percept occasioned by nothing external and substantial. (Hallucinations are rare; if they were not, we would not have a practical or pragmatic existence.)

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POSTULATES

Objectivity

1. There are entities and occurrences not reducible to my awareness of them.

2. Everything that exists can be assigned to a point or positive volume of space.

3. Everything that exists can be assigned to a point or positive interval of time.

4. Time passes.

5. Time must be divisible into instants which can be tallied as points on a straight line, i.e. one spatial dimension. [Time is demanded to have a linear geometry: a paradox which surfaces already in the postulates.]

6. There are entities which persist.

7. There is motion.

8. There is perceptible qualitative change.

9. The universe’s past, present, and future are all real always. [Otherwise the universe’s existence would be subjective! But then time is atemporal—it is frozen!]

Identity

10. An entity is itself; it cannot be something other than and distinct from itself. [A is not not-A.]

11. If an (identified) entity is finitely composite, consisting of a finite number of parts, then there is always a last part whose addition “completes” the entity, i.e. produces the entity, i.e. brings the entity to its identity.

Quantity

12. No entity (or occurrence) can be infinitely vast. [This may be a sectarian tenet, finitism, but it is legitimate to consider it common sense, since an infinitely vast entity is not thinkable.]

13. The “size” of a whole is greater than the “size” of a proper part of that whole. [Interesting for naive mathematics but not used here.]

14. Infinitely many positive magnitudes cannot be serially cumulated in a finite time-interval with a finite result. [This is Zeno’s postulate, conceived as a companion to the Archimedian principle, which is still accepted. There is a hint of constructivism in the serial cumulation clause. Mathematics has long since rejected Zeno, going where common sense cannot follow.]

Causation

15. A given occurrence has a preceding ocurrence (the antecedent) which produces it: meaning that if the antecedent is repeated, it will be followed by a repeat of the given ocurrence. The given occurrence is called the consequent or the effect. The antecendent (the cause) may be decomposable into multiple concurrent ocurrences. “Every occurrence has a cause.”

16. Antecedent, and effect, must be proportionate. (Magical causation is ruled out by common sense because it is unreliable, to say the least. What the stage magician does is indeed not as it appears; it is appearance masking what occurs objectively.)

17. An occurrence always has an immediate cause which immediately precedes it. If we identify a temporally remote antecedent, there is always a chain of causes from that antecedent to the effect. Causation does not transmit magically through time. A regress of causes is guaranteed.

18. Entities cannot vanish, or appear out of nothing. (In physics, conservation of matter.)

19. An effect cannot be its own antecedent.

Possibility

20. If the antecedent of an occurrence were different, the consequent could be different.

21. When I wish one of several possible acts, I could have wished a different one. (See below.)

Subjectivity

22. What we call a living alert person is aware. He or she self-consciously observes; he or she imagines; he or she wishes; he or she espouses propositions. (“Observes”—as opposed to coded registrations of events by a mechanical artifice—as opposed to the geological “record,” or tree rings.)

23. I observe extended motion or change without jumps, arriving at a future “Now” in the course of making the observation. That guarantees that my awareness has identity over a positive time-interval. My awareness persists, occupies definite stretches of time.

[Perhaps the explanation is that “observation” includes the continuous transfer of perception to memory, so that I judge the motion at its end by my uninterrupted recent memory. But if we try to be deeply analytical here, we leave common sense.]

24. From the vantage-point of the present, I do not perceive any future. (I may imagine it.)

25. From the vantage-point of the present, I do not perceive any past. (I may “remember” it; memory is not self-validating.)

26. Temporally, my perceptions are “Nows.” An instant is singled out by my awareness, but it instantaneously ceases to be the same instant. [A paradox which surfaces already in the postulates.]

27. Given that time passes, it is possible for me to arrive at a later time (a time which is future from the vantage-point of the present).

28. It is not possible for me to arrive at an earlier time. I cannot return to the past.

29. I cannot traverse time at different speeds.

30. Given that time passes and that I arrive at future “Nows,” I can judge (not perceive) the size of time-intervals in periods of continuous awareness with some reliability.

31. My awareness cannot be assigned to any spatial point or positive volume. [A paradox which surfaces already in the postulates.]

32. Each human’s awareness and body are anchored together. E.g. my wish can actuate my body, but not another’s body.

Freedom

33. For some of my acts, the act is one of several possible to me, and it occurs because I wish it. The act would not have occurred without my wish. So the act has a source outside causation. Inasmuch as my act would not happen without my wish, which is outside causation, the act is responsible. My responsibility consists in realizing my wish as an act.

Then my act has as its source not only whatever material cause is involved (the repeatable proportionate material antecedent), but my wish, which is not proportionate. [All this is a paradox which surfaces already in the postulates.]

34. My wishes cannot be impersonally caused; if they were, then there would be no responsibility.

Mind

35. The stretches of time occupied by my awareness are interrupted by stretches of unconsciousness.

36. I cannot subjectively judge the duration of my most recent period of unconsciousness with any accuracy.

37. A person’s awareness, anchored to a definite body, comprises a personal identity even though interrupted by periods of unconsciousness. Labelled the person’s mind. A mind persists throughout the individual’s life (more or less).

38. My awareness is one phenomenon in a multiplicity of phenomena.

39. I do not remember time before my birth.

40. My mind does not arrive at any time after my death.

41. Approximately, my mind’s final instant is concurrent with my body’s death. (The exceptions to this approximate concurrence of vanishing of mind with bodily death comprise fascinating details which common sense has not caught up with. Life after death is not recognized here.) Thus, my mind as an entity not only has location in time but has definitive bounds in time.

42. As there are multiple humans, there are multiple minds, each uniquely associated with a human.

43. I cannot perceive another human’s mind. (No matter what claims are made for communcation via words or comportment.)

44. There is language. [Facial expressions and postures also evince mental states, for whatever that is worth.]

45. Each mind possesses some knowledge in the form of self-conscious observations and espoused propositions which are veridical. [That is a deep claim of common sense: “if every proposition we espoused were false, we would not survive.” We pass to a different level if we realize that this common-sense truism is not compelling; the beliefs which are employed practically do not deserve to be called truths, and this insight is extremely non-trivial.]

46. An awareness (not to say a mind) does not know everything; it is finite. (I am always ignorant. Correlative to the existence of false propositions, I can be deceived.)

Objectivity of subjectivity

47. Knowledge is limited to our finite awareness by definition. (So claims for states of affairs that are unknowable in principle are problematic not only relative to credibility, but relative to meaning. The circle closes and our subjectivity confines objectivity.)

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PARADOXES

1. A uniformly expanding arc moves in an infinity of different directions at once. The ony way we can reach elements which are moving in a single direction at once is to subdivide the arc into points.

To discern translational motion of a non-vaporous entity requires (in general) that the entity be divisible into points. Each spatial point would then be occupied by a portion of the substance comprising the entity: the point-mass.

We are compelled to admit points as material; this will be found to be a dangerous result.

2. It is impossible to form a positive-length continuum by “stringing” points serially. For points cannot be contiguous (touching at their boundaries) without coinciding completely, since they are indivisibly small. And they cannot be non-contiguousy collinear, since if they are non-contiguous, a positive gap wlll intervene between them.

3. We found in 1 that every real entity, which must occupy a positive portion of space, must be decomposible geometrically. If everything real is decomposible, then it is decomposible without end, and we may conceptually carry the process of decomposition all the way to unextended points. But then the entity is an aggregate of elements which are unextended points spatially. It is impossible to form an extended continuum, a continuum of positive length, by aggregating unextended points. Because a sum of zero-lengths can only equal zero. [Please don’t bother to tell me that advanced mathematics grapples with this.]

4. We saw that points of a continuum cannot be contiguous. We have arrived at non-contiguous collinear points. But no matter how tempting it is to imagine that there is a positive gap between them, common sense already harbors a mystery which does not permit that. There must be limitless divisibility. If two distinct points are elements, then there is a point between them which is also an element. That is a dangerous conclusion. To give it a label we have to go outside common sense and say that points are asymptotically close. Non-contiguous collinear points without positive gaps must come about somehow. (Not by being strung together! To rationalize this is the history of mathematics, from the Greeks to the Souslin line.) Asymptotic closeness violates Postulate 14.

Objective time

5. We have Postulate 5, time-instants can be tallied as points on a straight line. But then there can be no positive time-extent. Because a sum of zero-lengths can only equal zero. Thus, ¬ Postulate 4.

6. A positive time-extent ultimately requires asymptotic closeness of time-instants. ¬ Postulate 14. Thus, ¬ Postulate 4.

7. Two cases of an entity’s persistence in time.

a. The entity vanishes at every instant, to be replaced by an entity which we please to identify with the vanished entity. ¬ Postulate 17.

b. Alternately, the entity persists continuously. Then, like time, the entity has “instants.” (To be clear that we are not merely talking about points of time, let us say moments.) The entity is endlessly divisible into moments. If a moment of the entity is defined by its time-instant, there is no way of divorcing the moment from that instant. The moment of the inert entity cannot transfer its identity to a succeeding moment. ¬ Postulate 6.

8. Further cases of an entity’s persistence in time.

a. An entity’s persistence cannot consist in a moment being replaced by another moment at a contiguous time, because the two times would coincide and there would be no persistence. But persistence cannot consist in replacement of a moment by a moment at a “next” time which is non-contiguous. Because then the former moment would occupy all the instants, between the given instants, which divisibility guarantees—contrary to definition.

b. Given that a persisting entity must have moments, its endless division into moments encounters the absurdities of summation to zero and asymptotic closeness.

¬ Postulate 6.

Motion

9. Motion occurs either in discrete steps or continuously. If motion occurs in discrete steps, then the moving element vanishes in one place and reappears in a separate place with no intervening process or phase. ¬ Postulate 18.

10. If motion is continuous, then given any positive interval of time, the moving element cannot be assigned a unique location during that interval. ¬ Postulate 2. The moving element is locationless, non-spatial, while it is moving—unless we subdivide time into point instants at which the element can be assigned unique locations. But since, at a given instant, the element has definite location in a volume equal to its volume, it is at rest. It cannot move where it is, because it is at rest there. It cannot move where it is not, because it does not even exist there.

What is properly called motion cannot be produced by conjoining an element at rest here with the element at rest there.

¬ Postulate 7.

11. We have shown (1) that it is necessary to postulate point elements of things in order to rationalize motion. But if a substance (a weight) is defined by its assignment to (identification with) a geometric point, there will be no way of distinguishing it from that point. The point mass cannot preserve its identity while moving from one point location to another. ¬ Postulate 7.

12. Assume that the moving element traverses instants in time. The notion of time-instants aggregated to form a positive time-extent encounters the absurdities of summation to zero and asymptotic closeness (as we saw). Ultimately, ¬ Postulate 7.

13. Let us make the issue of Postulate 14 explicit. In order to travel to a goal or to catch up with a moving object, one would have to traverse an infinity of positive distances (each half as long as the preceding one, say) in a finite time with a finite result. ¬ Postulate 14. Thus, ¬ Postulate 7.

14. Suppose we try to explain persistence of the universe1. Say that in the transition from past to present, or present to future, the universe vanishes into nothing, and appears anew out of nothing, at every instant. ¬ Postulate 18.

Change

15. Change: now we are simply hammering a paradox implicit in the definition. Change means that one and the same entity is constituted by two mutually exclusive totalities. ¬ Postulate 10. Or else: one totality vanishes into nothing, to be replaced by another totality which appears out of nothing. ¬ Postulate 18. thus, ¬ Postulate 8.

16. When an entity changes, how can the two totalities or “phases” constituting the entity be mutually exclusive without annulling the entity? The mutually exclusive totalities are assigned to different times. But time encounters the absurdities of summation to zero and asymptotic closeness (as we saw).

Temporality of causation

17. Change of an entity can be conceived as the causal relation of antecedent and effect. But temporality is again an issue.

a. One occurrence cannot be the immediate cause of another unless the occurrences are temporally contiguous. But we are forced to acknowledge the division of occurrences into temporal point-occurrences. Thus the cause of each occurrence is the prior occurrence contiguous to it. But if point-occurrences are contiguous, they must coincide completely. (They are indivisibly transitory.) That leaves each point-occurrence as its own cause. ¬ Postulate 19.

b. Does causation consist in the effect being produced by a prior occurrence which is strictly separate from the effect in time? ¬ Postulate 17. To reason another way, if we insist that an occurrence separate from the effect is its antecedent, then the occurrence must occupy all the time-instants between it and the effect—which immediately nullifies the assumption.

18. Anything that is not a heap of sand cannot be turned into a heap of sand by adding one grain of sand to it. ¬ Postulate 11 on identity. [This also aligns peculiarly with the definition of change: when a few grains are “changed” into a heap of sand by the successive addition of grains. Nothing but the number becomes different.]

19. If the universe1 is limited in space, then it is surrounded by an unlimited void space, which is impossible. The overall result: ¬ Postuate 12 (finiteness).

20. If the universe2 had a first instant, then it appeared from nothing. ¬ Postulate 18. The alternative is that the past is infinite. Now there is a regress of causes. This regress will be infinite. Causation falls into a paradox of an infinity of “elements of positive magnitude” in succession. (We cannot sort out the succession in the infinite past. No cause is first.)

Possibility

21. Existence as a predicate. Let me imagine that there is an existing one million dollars in my bank account. Then the existent one million dollars must indeed exist in order to satisfy the definition. Therefore there is indeed one million dollars in my bank account. Absurd.

22. Non-actuality/impossibility. Let us grant that there could possibly be a large sum in my bank account. There is a possible large sum in my bank account. Then that sum must actually be in my bank account, for if it assuredly is not, then it cannot be possible for it to be in my bank account.

23. We say that Cerberus does not exist and the Pegasus does not exist. But if they do not exist, then it is impossible to distinguish them from each other (to distinguish two non-existents or nothings), and it is impossible for their names to have meanings distinguished from each other (or to have meanings at all). Thus, nothing not actual can have an identity.

Subjectivity

24. My awareness, not to say my mind, persists in a definite stretch of time. All the same, it cannot be assigned to a point, or volume, of space. ¬ Postulate 2. [It has an anchor which it does not physically fill: my body.]

25. When the passage of time is invoked as an explanation, time is conceived as if it were one of the spatial dimensions, extending linearly in past and future with every point accessible always. This conception is contradicted by our experience of time.

26. Objective and subjective time clash immediately, because objective times are all real always, but the subjective temporal element, “Now,” somehow comes and goes, or changes. To spell it out. “Now” cannot be always the same, but neither can there be many different Nows.

a. If all Nows are the same, then what happened the day I was born is co-temporal with today.

b. If Nows are different, in which Now did the preceding Now case to exist and what Now replaced it? i) Now1 cannot be replaced by a contiguous Now2, because points cannot be continguous without coinciding. ii) If Now1 ceases to exist at a non-contiguous Now2, then it is co-temporal with the Nows, between the separate Now1 and Now2, which are guaranteed by divisibility.

¬ Postulate 4.

27. I observe extended motion or change without jumps, arriving at a future “Now” in the course of making the observation. And yet:

Temporally, my perceptions are Nows, no more and no less. Otherwise I would know time that does not pass, frozen time. [So my observation of motion or change is non-perceptual. This analytical conclusion exceeds common sense.]

28. Again. We postulate that awareness can register motion. (Postulate 23.) But even transient motion is not an instant. So awareness registers something that is not instantaneous. Then “Now” need not be instantaneous. ¬ Postulate 26.

29. Postulate 23 told us that my awareness has persisting identity. That rules out vanishing of my awareness and its instantaneous replacement by an other.

Yet, as it persists, my awareness has instants. Then awareness-instants separated by a positive time-interval must have an awareness-instant between them. My awareness is decomposible into instants without end. But if an instant of awareness is defined by its time-instant, then there is no way of divorcing it from that instant. The instant of awareness cannot transfer its identity to a succeeding instant of awareness.

30. Given that the instants of my awareness are temporally punctiform, they cannot form a temporal continuum. As before:

i. If awareness-instants are contiguous, they must coincide completely, since they are indivisibly transitory. Then there is no persistence.

ii. If awareness-instants are separated, there is always an awareness-instant between two separate awareness-instants. Then my awareness as a temporal continuum encounters the absurdity of asymptotic closeness. ¬ Postulate 14.

31. We are given that the past, present, and future of Universe1 are all real always. However, we have a past which we can remember but not reach, and a predecided future which we may (or may not!) reach but cannot know (exept by guessing). The only part of time which we occupy as perceiver is the infinitely small part which is the present. The objective past and future are in the general case not known, and may be unknowable in principle. Postulate 47 militates against Postulate 9 [Postulate 4 relative to the Universe’s lifespan].

Freedom

32. Now we are simply hammering a paradox implicit in Postulate 34.

My wish, which commences a responsible act, has to be “free,” spontaneous or uncaused. Then my mind is an entity not governed by universal causation. ¬ Postulate 15.

33. To expand on the preceding, my responsible act, which is physical or material, is not exclusively an effect of proportionate antecedents. (My free wish is not a proportionate antecedent.) ¬ Postulate 16.

34. For once in this manuscript, we will examine a “sophisticated” attempt to dispose of a difficulty—here, causation versus free will. It was submitted that causation applies in the empirical realm, while free will obtains in a transcendent realm out of time. That is silly, given that we continually demand responsible choices, localized in time, from each other. Free will cannot be rationalized by making it atemporal and ethereal.

35. Dogmatic objections cannot dispose of the way we know time, and of our freedom, so easily. For if the future is real (Postulate 9), then constitutive definitions are superfluous. There is no need for me to make an appointment with you, since whether you will show up is already set in stone. (And certainly is not going to be affected by a wish, which cannot be a proportionate cause.) This is a stunning rebuff to objectivity, even more urgent than the doctrine of responsibility.

Other minds

36. If many minds exist, then there are entities, capable of making observations, which are unobservable for each other in principle. ¬ Postulate 47. But this defeat for other minds is untenable. To wit:

a. If one denies that many minds exist, whom (or what) is one trying to convince? There is no way in which the notion of many minds could have arisen if it were not true.

b. If many minds are denied, then another person’s statemtents about his or her mental life are nonsensical. But if his statements about his or her mental life are nonsensical, then his or her statements about his or her bodily sensations are equally nonsensical, and his or her observation-statements about external phenomena are nonsensical.

c. If the only “observations” are those made by mechanical instruments or the geological record or tree rings, then there are no self-conscious observations and there is no knowledge. ¬ Postulate 19.

37. We must suppose that there is a reality not reducible to our awareness of it, and that our awareness is one phenomenon in a multiplicity of phenomena. (Compare the postuates of objectivity.) But this supposition encounters an unexpected difficulty. It requires our knowledge to rise above all human awareness so as to view ourselves in the context of everything outside ourselves. But knowledge is limited to our awareness by definition. This paradox was already evident in the statement following Postulate 47.

38. A complete mechanical description of a human would have its validity depend on whether he or she believed it, because whether he or she believes it is a part of the whole that the description must account for. Thus, the description is not valid independently of whether he or she knows of it or believes it. (Donald McKay. Further exploration of Postulate 47.)