Hume's Two Errors

(Or The Case of Mistaken Identity)

by Philip McPherson Rudisill

 Composed sometime before January 31, 2000

Contents

Appendices

Introduction

Occasionally our eyes play tricks on us, like when we think we see a roach on the kitchen table and, upon closer inspection, we find that it was merely a raisin.* We breathe a sigh of relief. sweep the raisin (spilled out from our breakfast cereal, perhaps) into the trash and go on about our business. It was an all too common case of mistaken identity.

[* But which does in fact resemble a roach in size and color; and for which reason, along with the fact of some recent concern about real roaches, perhaps, we say we understand this phenomenon of mistaken identity, ascribing it, as we do, to an associative tendency of the mind; and which (explanation) is satisfactory for every adult human being, ... except Immanuel Kant, that is.]

Upon a more circumspect consideration of this matter, that so-call nearer look that characterizes him who will be philosopher (or scientific in thinking), we realize that we were actually a bit precipitous in our conclusion, and that there is actually another, alternative explanation for what we saw, namely: that a roach was actually and physically present, and that we saw it, and that it then suddenly changed into a raisin before our very eyes (or while we blinked), and indeed perhaps even did so deliberately (for all we might know) in order to remain camouflaged to our view.* This would not be unlike stories we hear of the vampires, the legendary gougers of human blood, and who are able to metamorphose at will into wolves, bats, and even smoke, and which many people assert to be true, and without batting an eye.

[* This is not the place to delve into the matter, but I cannot help but think that the artistry of magicians cannot be appreciated until the tricks that they perform are first understood to be impossible in the way that they appear. It is, I suppose, like the story of Peter Pan and the episode of his lost shadow which is discovered rolled up in a chest-of-drawers, and sewn back on to his feet. Little children do not find this amusing, but rather, once they understand Peter's predicament, greet the retrieval with great relief, for who would want to be without one's shadow!?**]

[** From such considerations, by the way, the empiricists draw the wrong conclusion, for they reason that the only way that we come to think that the shadow cannot be separated from the feet of little Mr. Pan is because we get used to our own shadows' presence. But they forget their own experiments in trying to escape their own shadow (or their mirror reflection), which were a priori prompted and conducted for the express purpose of making this very determination.]

But most of us do not subscribe to this (albeit more intuitive) explanation, where we take perception for reality, but rather opt for the story that the raisin we saw was really there the whole time, only it "looked like" it was a roach when sighted at first glance and perhaps out of the corner of our eye. Now if this story is true, the story of the eye being "trickable", then that would mean that there is a real object, in this case the raisin (or the world whereof the raisin is a part), which remains the same whether we look at it or not, and that any recognition of anything other than that object would denote a problem in our sighting, the so-called trick of the eye alluded to above.

Traditionally two distinct schools of thought have been utilized to explain the knowledge that we have of objects (including the knowledge that we have of ourselves, like our penchant for mistaken identity): there is the empiricist school, according to which our knowledge arises from the exposure we have with objects, and that all we do is to abbreviate (by means of words) and refine what we have observed; and then there is the rationalist school, according to which we have a preceding (a priori) knowledge of these objects and merely search them out and discover them in and among the jumble and jungle of the sense impressions.* **

[*The marvelous story/ruse of Bishop Berkeley to the effect that there are no objects to be discovered [other than spirits (which are basically two, ourselves and God)] and that what we spy as the material universe is merely an ordering of perceptions, a situation which can be liken to people attached to virtual reality machines which are coordinated by a single programmer (God); this story, although unusually novel (and even curiously so), belongs to the empiricist school, and in fact might be called the fantastic epitome of that school, while Hume's more measured story might be called the realistic epitome of the same school]

[** See all the appendix on an Alternative Science for a consideration of how the rationalist might deal with the split finger syndrome.]

Both of these two schools sought to explain the knowledge that we have of the objects of our recognitions, both empirical objects, like tables and chairs, and also mathematical objects like one and two. And each saw insurmountable problems in the theory of the other; and each was able to defend his own position by attacking that of the other, for if there are only the two possible courses, then the refutation of the one proves the validity of the other.*

[*That logic of the excluded middle held until Kant decided that there had to be another, middle way, and set out to find it. The first stab was his Inaugural Dissertation, and his final solution was the Critique of Pure Reason.]

The common ground of both schools was the existence of the object of experience and the non-existence of the object of mathematics, i.e., that mathematics, and especially arithmetic, were merely an intellectual concoction, like a game, which posited ideas (numbers) which could be expressed in various ways (via numerals). And it was precisely here (with these presuppositions with regard to the object) that Immanuel Kant finally was able to discern a solution to the problem, for he decided to take the opposite tack and to assume 1. there were no object of experience (at least not given as such), and to assume in contrast 2. that there were an object of mathematics. By taking this road, the road less (or actually never) traveled,* he was able to devise a story into which the stories of both the empiricists and the rationalists would fit and might be derived (explaining the shortcomings of both, and not merely negatively, i.e., by reference to what was wrong in the other side), and one which explained the existence and validity of science, mathematical as well as empirical, and, at the same time, was compatible with the foundational (moral) premises of religion.

[* In an appendix to this essay, I will (seek to) demonstrate that Kant was actually very much influenced by Berkeley's conception of the world, though without subscribing to its more fantastic element, i.e., the absence of any object whatsoever.]

But it is somewhat misleading to say that Kant rejected the notion of a fixed object of experience,which both the rationalists and the empiricist assumed. To convey his thinking in this regard and to find a more precise expression, we will now consider this matter more closely.

The Object of Experience

The first insight into the problem of the relationship of the mind to object in the form called knowledge or recognition (Erkenntnis) came to Immanuel Kant, I believe, when in a reflective mode one morning he happened to notice that his finger which he had unconsciously brought to his nose had split into two, distinct, ghost or spectral fingers, and that he could, easily and just as surely as all humans had discovered at sometime in their childhood, make either of the two disappear by closing one of his eyes, and then make the other disappear by (only) closing the other eye.

At this point, I think, a light went on for him and it became suddenly extremely simple. On the retina of the eye are two sorts of objects: we have images and we have specters. Now the image and the specter are identical subjectively, i.e., within the retina there is no difference at all. The difference arises solely by virtue of the fact that the image presupposes an object in space to which it corresponds, while for the specter there is no such object and thus there is no like correspondence.

The epitome of this situation used by Kant (to convey the sense of this discovery to others)* was the rain and the rainbow. The rain on the retina of the eye reflected an object, called the rain, out before us in space; and for which reason the rain on the retina is called an image of the real rain in space. The rainbow, on the other hand, does not reflect an object which is located in that rain at all, but rather arises only within the eye itself, and for which reason we do not call it an image (for there is nothing that it images), but rather a specter or mirage or even ghost. (Erscheinung).

[* I do not mean to imply that Kant discovered the difference between images and specters empirically (for Newton had given the definitive answer on this earlier with his work on optics); but only that he realized that this distinction held the key aspect of the solution of the problem which had so long divided empiricists and rationalists in the realm of philosophy.]

Now since, subjectively speaking, i.e., with respect to the retina and the physiological make up of the human in general, there is absolutely no difference between the image of the rain and the specter of the rainbow, the question must arise as to how it could arise that we might ever have thought that there were such a difference objectively speaking? This is the great question that Kant now has as his task to answer. Neither the rationalist can explain it (for they have no concept of image which were distinguished from specter), nor can the empiricist explain it (for they are wedded to empirical data, and there there is no difference, empirically speaking, between the constitution of image and specter).

Now since both the rationalists and the empiricists are derailed by this question, the only possible solution is to decide how anyone would have come to a conclusion which is neither given a priori as such, nor suggested in any way by the empirical data. Fundamentally then the question resolves itself into this: how does any one originally, merely as a human being, come up with the distinction between objective and subjective in the first place?

Kant's solution stands directly between that of the rationalists and the empiricists. Like the rationalists (and unlike the empiricists) he posits an a priori knowledge, but then like the empiricists (and unlike the rationalists) he posits no a priori knowledge of objects. In other words Kant has the human being conceiving of a world of order and lawfulness, a world which makes sense, and then, with that conception in mind (which is generally denominated as nature), the human begins to make incidental, even inadvertent sightings which suggest or hint of pattern and regularity, and then consciously replicates these sightings to make sure.* Once the pattern as been established (a perception), then the human dreams up a specific object (based on the general notion that things make sense) such that what is sighted as the pattern can be necessitated. The object so conceived must not only explain the pattern, but it must also fit together in a coherent, expanding and open-ended and reciprocal system with all other objects.

[*This replication of the observation is itself a product of the categorical structure of the human mind, at least to the extent that this structure is the mental expression of order and regularity that is expressed sensitively in the pattern.]

This object of experience then is both a priori (for it is given before experience as the form of experience, namely with respect to its regularity and order) and also synthetic, (for it is composed [and even fashioned] of elements given in experience). It is by virtue of this object of experience that we are able to fathom the difference between the split finger as an image, i.e., that the finger actually splits, and the split finger as a specter, i.e., that the finger does not split. The object of experience in this case is actually the eye itself, i.e., we learn that the eye creates its own objects which, if taken for reality, result in illusions.

Kant discourses on this subject in a very similar vein in the Aesthetic.8.III.3 and especially in the footnote, which I quote here:

"The predicates of specter can be attributed to the objekt itself in relationship to our senses, e.g., the red color or the scent to the rose; but the illusion can never be attributed as a predicate to the object, and precisely because it would attribute to the objekt on its own what only pertains to it in relationship to the senses or generally to the subject, e.g., the two handles which some people first ascribed to Saturn.* ** What cannot be encountered with the objekt on its own, but always in its relationship to the subject, is specter and therefore the predicates of space and time are quite properly applied to to the objects of the senses, and in this there is no illusion.*** On the other hand, if I ascribe redness to the rose on its own, or handles to Saturn, or extension to all external objects on their own without considering a determined relationship of these objects to the subject and limiting my judgment to that, then only does illusion arise."@

[* The first sightings of Saturn were from such an angle that the rings looked like two handles.]

[** One could say as easily that while I cannot deny that I see two ghost-like fingers before my eyes when my finger approaches my nose, I cannot say that the finger itself actually splits, but merely that I see a split finger.]

[***There is no illusion in sighting the split finger (for this is a necessary result of sighting things in space with two eyes, and then unifying the two visions within the brain). Likewise there is no illusion when I say that as I draw my finger closer to my eye the split becomes more pronounced, or saying, in another context, that first I saw a flash and only later hear the roar (which I later objectify by saying that the flash and the roar were "of the cannon" and that the flash and the sound actually were simultaneous, only that light travels through space faster than the sound). The illusion arises when I saw that the finger splits without keeping in mind that I am speaking of a specter, i.e., a product of human vision.]

[@ Or if I said that the finger itself obviously splits, and do not speak of its spatial and temporal relationship to me, then I live in the illusion that the finger actually does split, even as I would live in the illusion of objects getting smaller on their own if I did not keep in mind that the smallness is correlated with the distance of the object from me.]

Now a rather formal exposition of the process taken in making the distinction between the objective and the subjective worlds is given in the essay on Circles in the Air. Here I want merely to go through an example of how someone might approach the "problem" of the split finger. I will take the role of an infant or young child who has become bored with the stimulus of the senses (though possessed the language of the adult).

I just noticed (or thought I did) that the finger split into two ghost fingers, but when I look at the finger more closely it is only one.* But there! it happens again! the finger split! but then when I look directly at it again,**it is only one. But now, when I bring my finger up close to my eye and look beyond it I notice that it does indeed split into two, very distinct ghost fingers. This is very interesting.***

[* Physically I am holding my index finger up perpendicular to the floor about about 1/2 meter before my nose. I look "through" it to the background and see the finger split. I look directly at the finger and see it rejoin (and the background split).]

[** The hint of the spectral nature is given in this clause "when I look directly at it", for if this is left out the suggestion would be that it is the finger itself which splits and rejoins.]

[*** The interest is really more an intrigue, i.e., the attraction of a problem or a mystery. This interest is entirely different from the pleasure that might be taken in the sight of the two fingers themselves, or in the pleasure associated with the finger on its own as a play thing (where you can stick it into food and then stick it into your mouth), for this interest is entirely intellectual, i.e., constitutes a puzzle, i.e., a pattern (or break in pattern) which is unexplained. The only basis for this intrigue for this can be assumed to be the Transcendental Object = X, i.e., the general form of a unified consciousness, which is expressed specifically by the categories of pure thought as the intellectual means of unifying a multiplicity in general. At this point we have our first perception, i.e., the fact that the finger does indeed split when it is drawn near to the eye. This perception then is a function of the experiment and the observation and the subjective conditions of such experimentation.]

The experimentation continues with the further observation that the ghost fingers vanish, but in tandem, as the eyes are successively shut and opened. This is further sighted against a splitting backdrop of further objects. The final recognition is that the split finger is really only a specter and what is "really there" is the image of the finger in each eye and which (two sightings) are blended in the brain, but which does not exist that way on their own.

This may be the place for a short discourse on our understanding of space, which is greatly facilitated by this examination of an experiment which is necessary for every human being in order to make sense of the split finger phenomenon.

When I see my finger split at my nose, I explain this effect (as I call it)* by reference to a sighting of my two eyes in space, i.e., each of my eyes is focused on the same object, a single finger, and what I normally see as a single finger is actually the blending of the finger sighted by each of the eyes into an amalgamation that I call the finger. Thus the real trick is that the single finger that I see at a distance is really this amalgamation and not singular at all, but always two. The truth of this assertion can be established by sighting at the finger before a diverse background further remove and notice how the back ground splits when the fingers join, and how the fingers split when the backgrounds join.

[* And we might want to keep in mind that this is not the only explanation. In an appendix entitled "Alternative Science" we explore the possibility of a science based on the fact that the finger does in fact split.]

Now in order for this explanation to work, I must conceive of a single object, the finger, such then that the ghost or spectral fingers can be seen to be an image of this object. But that object cannot merely be dreamed up gratis and accepted as such, which would be a wild story indeed, but rather must be found to actually occupy space.

But the space that it occupies, this real space apart from me in which objects can exist at the same time as other objects, this space I do not actually see as such; for if the space in which I see the divided ghost fingers were really the space around me as an object, then that would mean that the fingers really did split and that I am not dealing with any illusion when I assert that, e.g., "say what you want to, I can see two fingers touching my nose."

But on the other hand, and to make the expression a bit equivocal, this is the only space that I have any access to, i.e., the space of my vision. Now what actually occurs is this: I see the finger literally in space before my eyes, but I also know that that space in which the split finger appears is really only within me as my way of looking at things, and so I realize that the space I actually see, while real enough and while showing objects really apart from me, that space is merely a sighing within space and indeed really is that space, but only from a particular point of view or a particular vantage point within that space. And for this reason I do not hesitate to say: I actually do in fact see real 3-D objects in a real 3-D space before my eyes, while continuing to admit that what I see is entirely within me and not apart from me in the way that it appears, .e.g, with split fingers.

In summary then, with regard to the object of experience, we have a preceding envisioned space (and time) and we are able to see all (external) objects in this framework and in this context. Space and time are are environments, as it were, for patterns to arise. The human, by virtue of the Transcendental Object = X (the form of a unified consciousness), is intrigued with patterns and seeks to necessitate them. Then a second look is undertaken, in pursuit of this necessitation, and this second, certifying look results in the establishment of the fact of the pattern, and this is called perception; hence Kant's powerful assertion that perception is a function of the category, at least with regard to the prompt to the so-called second, certifying look (which takes place only with empirical data).*

[*All of this is presented in considerable in the essay on Circles which can be found on my home page.]

Hume picked up on pattern, of course (as an empiricist), but could not think it, for he despised the rationalists' story and for that reason also insisted that all objects came from the senses. Leibniz (the ultimate expression of rationalism) thought he understood the error of the empiricists, for they could never be sure of anything, where simple analysis told you that things had to be the way they were. But then he was unable to distinguish his left hand from his right (conceptually), or to distinguish a specter from an image (by reference to the object).

Kant sees that each is right, Hume with the patterns, and Leibniz with the object, but only where the object is merely a unification device (of the elements of the pattern) which is applicable in time and space, for it is there that we get the content of the stories, which is really what an object is in a way. This object is called the transcendental object = X, and it was essentially what we mean when we say that something is really_there. Like Hume's table is really there. And whatever might be its causes is really there, etc.

The solution then comes about in the provision of the object by the mind in order that a distinction can be made between image and specter. Now the provided object comes about in a mechanical sort of way by virtue of the structure of the human mind. The form of the human mind is two fold: there is a sensitive aspect and a logical aspect. The sensitive aspect has to do with sightings which take place in a context of time and space. The logical aspect has to do with thinking, and this takes place in a context of universal assertions and propositions. The conjuncture of the two is called a recognition (and we will see a similar conjuncture when we turn our attention to mathematics).

The Object of Mathematics

[Note as of 6/16/01: Two briefer and more concise treatments of this can be found at 7+5 = 12, and 7+5=12 again.]

Now we have seen how the mind provides an object of experience a priori and yet synthetically. The actual object recognized is, of course, empirical, but the form of that object is a priori. The table that Hume knows to be fixed and independent, and what he (very early) conceived in order that the fickle specters (Erscheinungen) might be demoted from things on their own to mere views and images of things, that table is the empirical object, and is the empirical manifestation of the Transcendental Object = X = nature (TO=X), which in turn is a priori and synthetic and the basis for making the observations and experiments such that the table then is recognized.

Hume's error here (with regard to experience) was in taking the objects of experience as given on their own. But this emphatic effect (of permanence and independence) exemplified by the objects is merely the result of the synthesis undertaken by virtue of the TO=X, and this Hume did not recognize. And so as far as he was concerned he was faced with pre-existing objects and, of course, had difficulty in figuring out how anyone (perceiving them externally) could know anything about them for sure.

Now in order to give the devil his due, but without granting any credence to the fabrications and metaphysical worlds dreamed up by the rationalists, Hume ascribed to mathematics a certitude that he denied to objects of experience (thinking always that these empirical objects existed as such on their own, and therefore that all knowledge of them was always problematical*). He did this by accounting all mathematical knowledge as sheer logical relationships which accorded with, and even proceeded always via, the law of contradiction, i.e., two expressions were considered as synonyms unless doing so resulted in a contradiction. For example 7+5 and 12 would be thought as synonyms unless this entailed a contradiction; and such a contradiction would occur, for example, if 8+5 were considered as the same as 7+5, etc.

[* He did not realize, as Kant did, that the objects were provided by the mind in its search for expression to the order and regularity which were a priori presupposed (for and in the specter) in order that perception of the object itself might occur--this is examined briefly above and in more detail in the essay on Circles in the Air on my home page.]

In this way Hume completed his scheme of human knowledge: probabilities with regard to objects of experience and certitude by means of the logic in mathematics.

But it is here also that Hume erred, and indeed stumbled at such a very simple step that 1. it is understandable that he might miss it, and 2. had he not missed it, he would have formulated his concept of experience entirely differently (and, as Kant himself stated, would have been the author of the Critique of Pure Reason). I shall seek to make this clear with an example involving my 1987 BMW model R65 motorbike.

I have been told that the R65 was a hybrid, a something assembled from parts for other bikes. I shall assume that this is true for the sake of this example; namely that every part in the R65 was originally designed for, and utilized in, another machine, be it a motorcycle, a lawn mower, an automobile, a boat or even a refrigerator. Furthermore, for ease of expression, I shall say that the R65 is made entirely of parts a, b and c.

Now I ask, "what do you have when you have an R65?" and I get a resounding "a+b+c!" but when I ask, "what do you have when you have a+b+c?" I get only "a+b+c" for sure, and otherwise only a "it depends." Now the first answer (to the second question), i.e., a+b+c, is hardly edifying (being a tautology), and we could have saved our breath. But the second answer is hardly what we would call definitive. But it is the only answer that is possible under the circumstances, for we really cannot tell whether or not we have an R65 or not.* To know for sure what dealing with an R65 we would simply have to take a look.

[*This might be even clearer if we say that we have a+b+c+d; for then it should be very clear that we might very well not have an R65 plus part d, consisting as it does of a+b+c (although this expansion of the components is really not necessary to establish the point that we cannot be sure from the components whether or not we have an R65 motorcycle).]

Thus to know that an R65 means a+b+c does not tell us at all that a+b+c means R65.*

[* Some objection to this thinking might arise from those logicians who think to include an ordering in the listing of the parts, so that to them, since an R65 is a+b+c, in order to be consistent, we are to think of the parts more as though they were b+c+a, or some such, and whereby they (the parts of the unassembled bike) are not perfectly equivalent (to the parts of the assembled bike) in that they are not in the same order. In this they appeal to (what I have come to call) the fundamental axiom of arithmetic, namely: if the elements of X are indistinguishable from the elements of Y respectively, i.e., the first element of X and the first element of Y are indistinguishable, etc., then X and Y are indistinguishable. But then we, for our part, appeal to Kant's own example of incongruent counter parts,** and consider two spherical triangles, ABC and AB'C, where B and B' are on different sides of the common base AC, and where AB, BC and CA are all unequal, and where AB and AB', and BC and B'C, are equal (and CA is, of course, common), and where BB' does not bisect CA; and find that the parts of each are equal, respectively, to the parts of the other, i.e., in the same order of assembly, and still the two triangles are as different as the left and right hands.]

[**Prolegomena to Any Future Metaphysics, Section 6 "How Is Pure Mathematics Possible?", subsection 13.]

And so it is universally! To know that having an 8 means having a 7+1 does not mean that by virtue of having a 7+1 I have an 8. And yet that is the only way that I can come to know logically that having a 7+5 is to have a 12 (to refer to Kant's famous example by which he deprecates the logician's assertion that heshe can know that 7+5=12 from an analysis alone of the terms involved); for the necessary procedure (logically) to come to that conclusion (that 7+1 is 8) is to derive a 4+1 from the 5 (which is proper) and manipulate it a bit to come up with a 7+1+4 (which may also be proper) and then to go from there to a 8+4 (and so on to 12). But it is now clear that this (derivation of an 8+4 from a 7+1+4) is an unwarranted jump, logically speaking; for otherwise I would be able to infer without possibility of error that having the BMW's a+b+c means that I have the R65, and that is simply not a valid conclusion, even though it may be true in fact (but which would call for a look-see).

Now the reason that Hume missed this insight, I suppose, is because it does happen to be true, without exception, that in mathematics having a 7+1 does in fact mean having an 8, and therefore that the equation of 7+5=12 is universally valid and necessarily true. But the reason that we recognize that the relationship between 7+5 and 12 is "=" (equivalence) and not merely "is" (predication) is not logical (as we saw in the preceding paragraph and its footnote), but real, i.e., in the same way that when I see a+b+c assembled in a certain way in space I know that I am seeing these components as an R65. But the certitude of the mathematics must be even beyond the certitude from that empirical sighting (for it were possible, I suppose, for there to be an hallucination of some sort with regard to the motorcycle).* **

[* As an extreme example, we could imagine that in the storage of the parts of the BMW factory a few parts happened to be stacked in such way that the semblance of an R65 arose, much as a face can be sighted in and amongst the textures and contours of a cloud.]

[** There is a similar situation here as there was in the case of the mistaken identity with regard to the raisin on the kitchen table alluded to at the beginning of this essay. It was only by virtue of the confidence of independence and uniform of the object were able to conceive of a misperception, i.e., a mistaken identity (for otherwise the raisin/roach would be considered to be a thing on its own and thus beyond the pale of human knowledge, and subject only to a calculus of probabilities). The same must hold true in counting in order for us to be able to conceive of a mistake in counting as opposed to a change in the objects being counted or in the uniformity of the count itself, as a logical contrivance. (I am not sure that this consideration has occurred to anyone before.)]

The solution to the problem of the validity of arithmetic is paradoxical. We are to speak universally, i.e., that any 7+1 is an 8;* and for this we turn to something which is entirely subjective, namely space; but within which we are able to demonstrate something objectively; for even though the space in which we sight things is entirely within us, and therefore is entirely subjective, nonetheless we are able to point out to others what we are referring to in that space by locating the object of discussion (Gegenstand) or reference and/or by tracing it out. The most emphatic example of this, I suppose, is the pantomimic circle traced out in mid air by the mime artist.**

[* Indeed we are to say that there is only a single 7+1, and that this is merely a concept without any object but merely an application; even though there is a pure, non-empirical object, namely space (and the warrant to which is the subject of this paragraph).]

[** A fuller explanation of this subjective/objective aspect of space is given above in the discussion of the object of experience.]

But the situation with regard to arithmetic is no different. I locate the units (objects) of arithmetic on the number line (like a row of dots, or along my fingers) and see immediately that unlike the objects of experience, I am not dealing with sensation but merely with locations, and therefore, by virtue of this sighting, I can tell that (unlike the R65 and a+b+c) 8 and 7+1 are merely synonyms which refer to the same point in space. It is in this way alone that I can distinguish the case of the a+b+c of the R65 from the 7+1 of the 8* and conclude that while the parts a+b+c may not signify an R65 the "parts" 7+1 do in fact signify an 8 and indeed universally and indeed such that the "is" of 7+1 is 8 can be replaced by the equivalence sign of "=".**

[*Indeed I can go further and see that the 7+5 (or 3+9, etc.) is merely a partition of 12, while 13-1, for example, is not; a distinction which is absolutely impossible in logical mathematics.]

[** I wonder if it is not by means of this mathematical equivalence (such as 7+1 and 8) that we first come to the real concept of a definition (as opposed to a mere assertion), i.e., a juxtaposition where the the elements can be substituted for the other. The proper reading of a definition such a 7+1=8 is actually of this wise: to say 7+1 is to say 8, and to say 8 is to say 7+1. It is surely in the contrast of the predication of an empirical concept, e.g., that an R65 has tires, with the pure concept, e.g., that 7+1 and 8 are synonyms, that the concept of definition has real meaning, as opposed to merely a logical one.]

I wish to utilize another example in order to more smoothly integrate the respective importance of, and the difference between, empirical and mathematical objects.

When I hear the sound "table" and look about to make sense of this sound, i.e., to turn it into a word, I come across a top and legs and formulate a concept of something which serves for holding things at a convenient height for, and accessibility to, humans. By means of this technique or procedure (a facet of human minds) I come to recognize the table (spectrally) and to see that the sound "table" is a word, i.e., refers to an object. I can then analyze the concept of table to derive the necessity of the top and legs, such that if someone tells me heshe has a table without a top, I can say in turn and (relatively) a priori, "then the top has been removed for some reason"* And thus top+legs can be derived from the concept without a passing glance at the object itself; and thus analytically.**

[* Which is the essential meaning of the empirical necessity which attaches to any concept.]

[** Keeping in mind that the concept itself was prompted synthetically and a priori, and only the specific result of the synthesis is empirical.]

But I cannot derive the concept of table from the concept of top and legs (as we have just established in the case of the R65 motorbike).*

[* For otherwise, if I saw a man carrying a small chest of drawers, since heshe has legs and it has a top, I could say that I saw a table along with some extraneous stuff like drawers, a belt, a couple of hands, etc., and all of which (extranii) might happen to be found on a table any way.]

Now the only difference between an 8 and a table, conceptually speaking, is that the 8 can be spotted universally in pure space (the number line) and found to be nothing more than the 7+1 (or 4+4, etc.), and for which reason we see how it is possible to speak a priori of the 7+1 and 8 (for they are one and the same, being merely diverse expressions of one and the same thing, i.e., an identity), but not of the top+legs, although synthetically in both cases. The table is an empirical recognition and therefore always somewhat problematical. The 7+1 and 8 identity is a pure recognition and therefore without question.

Therefore then (and rejoining now him in the CPR.Intro.V.1.) we see why Kant says that 7+5=12 is synthetic and a priori and how this is different from saying (in my own case) that a+b+c means an R65 or legs (or that top and legs mean a table), even though a peek at the object is necessary in both cases, the former being however a pure peek and the latter (two being) empirical.

Conclusion

We see now how Hume missed the boat. If he had understood that the validity of mathematics was dependent upon a look at the object, he would have had reason to suspect that the mind might also supply (conceptually) the object of experience such that the specters of the senses (and most especially the retinal objects) might be unified into views and images and effects of these (uniform and regular) objects, and that it were by means of this that the categories, such as causation, obtained their useful application and validity (as the form of unified consciousness). And he would have still been right to remain skeptical of the knowledge that we obtained about these objects, being empirically based, but not of the means of achieving to, and of recognizing, them.*

[* And Hume would also have resolved the doubts he expressed in the Enquiry. Sec.118 (back button must be used to return here) concerning how it was that he could tell that his table did not change size and shape as he moved closer to it and then further away fro it. This (and the ensuing section) present a fascinating picture of a man who has incredible insight and determination but who is unable to conceive of the solution to a problem which was staring him in the face. Kant insists that the reason was Hume's misconception of mathematics.]

So in both cases, empirical and mathematical, a sighting is necessary, and thus an object must be available for this sighting, and in both cases an object is provided by the mind for this very purpose, an empirical object (like the table or the R65 motorcycle or the single finger which splits) and also the pure object (like the pantomimic circle in the air or the arraying of numerals along the number line).

And thus it is that by means of an a priori provided and synthetically constructed object that not only the experience of mistaken identity* is gleaned from our exposure to the world, but also the recognition of the validity of logical mathematics.

[* Not to mention all sorts of other mistakes and distortions, such as the changing appearance of objects in space and time, acting (driving, for example) during periods of preoccupation, i.e., subconsciously, sleep and dreams, mistaking not hearing with no noise, etc.]

Appendices

Berkeley's Influence on Kant

In modern parlance Berkeley conceived of a world in which created spirits (humans) were hooked up to individual virtual reality machines and these machines were coordinated by God (the chief programmer) such that the impression of the supreme object of experience, the visible universe, was conveyed.

Kant rejected this notion, of course, for he knew of the existence of the material objects, just like Hume and Leibniz (and even Berkeley*) did. But he did come this close to Berkeley: in our endeavor to achieve to an objective world such that what we see and sense is merely a part of, and could be derived from, that objectively existing world, we conceive of a nature which is fixed and orderly and exists whether we are looking at the objects of that nature or not (which corresponds [a la Berkeley] to the array of perceptions in God's own supreme, virtual reality machine (the server, if you will), which encompasses all of the network machines as well as the server, and which is on all the time and where all possible perceptions can be located [existing all the time] and even those perceptions which will never been seen by a human (virtuality reality) machine, e.g., the back side of the moon in the year 1700). This is a function of the Transcendental Object = X, an arrangement and order of our minds, and by virtue of which we are able to grasp and recognize that what we see is merely a perception, and not a thing on its own (in contrast to Berkeley's system of idealism, i.e., where there were nothing in addition to the perception/sighting), but (continuing now with Kant's system) that there was this thing on its own which we did in fact have in sight; but since we sighted through our senses, this thing looked differently (in perception) than it did in reality, e.g., Hume's famous table which keep getting larger and smaller depending on how close he was to it.**

[* It is difficult to think that Berkeley actually believed in the truth of his story as rather in the belief that it could not be refuted, and therefore that it confounded the assertions of the materialists. Kant figured out the error in Berkeley's story, however, and reported it in the Aesthetic to Pure Reason (Section 8, Division III). Namely since Berkeley reasoned that things could not appear and disappear on their own, in contrast to the clear testimony of the senses, but rather that we were dealing solely with perceptions, it follows that the objects of the inner sense, e.g., the perceived manifestations of the soul, would have to be sheer perceptions also, and therefore Berkeley, to be consistent, needed to abolish his perceiving spirits just as much as he needed to abolish material bodies, and thus what we called the perceiver was himself merely a perception in the mind of God (but then even God would be come a perception!?), and not a thing on its own at all.]***

[** See the introductory paragraph to the Circles essay.]

[*** I am not yet entirely satisfied at this attempt to explain Kant's position here, and I anticipate further meditations before this becomes transparent in my mind. Essentially all of Kant's arguments are based on this premise: it is not possible to explain the explanation from within the explained system, e.g., Hume cannot explain how it is that he can distinguish a table from the image of the table since by his own system all he has is the testimony of the senses, and they are all of one coin, i.e., seated within us where the rainbow, for example, looks like it is really out there in the rain.]

A Speculation on the Change in Kant's Thinking from the Inaugural Dissertation to the Critique of Pure Reason, with Respect to the Object

I want to suggest that at the time of his Inaugural Dissertation pure space and pure time were (for Kant's scheme) abstractions from empirical sightings, and not yet positive envisagements on their own. But this (it seems to me) would have given an empirical order and validity to mathematics.* Kant somewhere realized that mathematics must be based on an object in order to give order and certitude to arithmetic, and at the same time could not be an empirical object. To accomplish this he dreamed up space and time as pure envisagements, positive on their own, as givens, i.e., as the way that we see objects, and not as an intuition from actual objects, which would have brought everything into question.

[* For it might have been possible to find a space in which 7 (units of that space) and 5 (additional such units) did not add to 12.]

It is for this reason that I think that while intuition is a proper term for the Dissertation, when we come to Pure Reason and Anschauung we need to think more in terms of envisagement. For surely the pantomimic circle located in mid air is not really there in order to be intuited (which would be a strange way of speaking), but rather is added synthetically and a priori via an envisagement such that it can be pointed out to, and discoursed on with, other people.

In sum: while in the Dissertation Kant was intuiting existing objects in space and time and merely regulating them via the intellect, in Pure Reason he was dreaming up objects which could be definitively sighted via the category-driven envisagement (and envisagement-prompted category) and thereby recognized as existing on its own independently of the viewer. And so while these sighting are very valid and the objects do in fact exist as such, they did not exist as such on their own independently of the human viewer, i.e., transcendentally speaking. And that is the teaching of Pure Reason.

Importance of Kant's Idealism for the Concept of Freedom

[This appendix is more of a sketch of a future essay and gives merely the gist of Kant's reconciliation of natural science and freedom which itself is given in the Third Antinomy of the Dialectic of Pure Reason.]

Since the object (of experience and even in general) is a priori and provided by the mind and not taken from the senses (except with regard to its empirical garb and as was established above under the heading of "Object of Experience"), it follows that we could actually think this object [= a thing on its own, i.e., what is really there] in a different way if we wanted to. But the opposite of necessity [which is what the object provides to the pictures (specters being the content of the pictures)], is freedom from all time and space constraints. We then, as long as we do not tamper with the structure of the picture (= contradict what we clearly recognize), we then remain perfectly free to say whatever else we wish about the object, and without seeking permission of any one. [Others will not listen to our story, however, if it is not logical; nor also not if contradicts the specter.] Therefore, for example, we can conceive of an steam engine which can think and talk, teddy bears that play with us, leaves that want to fly away (and only look like they have to fall), and what have you. And as long as none of this contradicts the specter we have committed no offense to the realm of science.

Now to our way of thinking (in our role as scientists) all of this is sheer fantasy (but which still does not contradict any science) and it is pointless to entertain it seriously, except for the sake of children's stories. But we (grown ups) are intellectual creatures and we are conscious of our capacity to deliberate and act in accordance with such deliberations. And therefore we can see a positive meaning to such otherwise inane babblings, i.e., we can now imagine ourselves engaged in serious deliberation with our own souls about some endeavor and proposed action, and even though, when we have finished and have undertaken whatever action we do, and can look back and ascribe it (as scientists) to our background and upbringing, etc., still, in another referral (as rational adults), we will also be conscious of having actually made a choice and that we are therefore free to act differently from the requirements of science, although we can never recognize this objectively such that it might be pointed out to another in the same way that we can point out the results of natural necessity as, for example, when we toss a stone up into the air and describe its fall.

And we can continue on in this way, with this thinking,* to the reconciliation of natural necessity with freedom, both referring to the same thing on its own, but the one as a thing on its own, and the other (the scientist) to the thing on its own meaning the object of experience (to which he in religion or morals or law, for example, is not limited). Thus they speak in different realms.

Summary: The point made is that we can think anything we wish and ascribe it to the object (including our own souls), considered as a thing on its own (and not specter), as long as we do not occasion any conflict with science. And this means that we can think that what people do (or what leaves do, for that matter),* they do freely. And this then opens the door to moral freedom where people are actually conscious of making a decision* which has moral implications, and this despite the fact that they and science may also so that their choices are all predetermined. This is a fascinating part of Kant's thinking, containing, as it does, the possibility of a reconciliation of moral religion (and law and freedom in general) and science.]

[* And where we then begin to leave out the possibility of leaves or animals and indeed even children acting in this way, and reserve our speech and assertions for people of will, i.e., a capacity to act in accordance with principles, most especially ourselves.]

An Alternative Science

I think it would be interesting to explore how each the empiricist and the rationalist would approach the split-finger syndrome on his own, without assistance from Immanuel Kant.

The empiricist would assert that sometimes there are a double (split) finger and sometimes a single finger. Heshe would then want to explain how it is that the finger(s) know when to join together into one, i.e., how they know that they are a certain distance from the eye (which is really a rational component in their thinking), for the pattern would be very clear. Also it would be necessary to calibrate the language of other persons, for sometimes when I see the split fingers, the others do not, but instead see only a single finger. And so we essentially have to explain the problem of how the same fingers can be split and not split at the same time. There would like have to be some appeal to some divine interference (again really a rationalist contribution). But the best bet would probably to take a Buddhist approach* and try to have a cup of nice hot tea instead, and not worry too much about whether the cup splits into two or three or even more, as long as the tea tastes good and enables us to enjoy a pleasant afternoon.

[* In my effort to comprehend the Buddhist mind, I think the solution they attain is finally not caring about such things as split fingers, and therefore not really being interested in all that to begin with. This, I think, is exemplified by the scene of the mad elephant charging through the village and falling dead at the feet of Gotama the (first?) Buddha who then turned to his companion (I have been told and while the village was going wild about the "miracle" which had just taken place) and suggested that they take another path since a dead elephant was blocking their chosen path; a very practical solution which is unencumbered with questions as to why or how the elephant happened to die at that moment and place.]

Looking now at the split finger from the rationalist point of view, we will easily come the conclusion that there is not a single finger at sometime and a double finger at another, but rather two fingers which only look like one when we are too far away to see accurately. Thus it is the distortion of the senses that makes it look like there is only one finger The solution is that each eye sees a different world and that these two worlds appear as one at a distance, but in fact are as different as night and day. And in the same way that each human coordinates these two worlds in the mind, God coordinates all worlds into a unity called existence.*

[* Leibniz goes on, as I understand him, to imagine that our own bodies, like all objects, are spectral on their own, as things, and that what seems to be a movement of our bodies through space is really merely a sequential "flashing" of the monads (points of view) which do in fact exist such that we appear to exist (as bodies) and to move in the same way that bodies seem to move across a television screen or objects move about on electric signs made up of a host of coordinated light bulbs. To justify this Leibniz had to ascribe a point of view to each monad then then also an ability to reflect all other monads in a sequence which, to our understanding, is called space and time, but which actually functions according to other laws. Thus each monad, for example, reflects the mind and soul of Philip, but will do so sequentially and in coordination with all others so that the soul and mind of Philip can always be located, but will "move" from monad to monad as each is programed to reflect Philip's soul and mind more vividly. The conception is mind boggling and wonderful and a creature of unsurpassed genius! Indeed perhaps the ultimate, metaphysical contrivance!?]

It may now be instructive to seek to contrast these two positions with that of Kant, as explanations of two phenomena, that of the split finger and that of the shadow.Thu we have three alternatives with regard to the split finger, namely:

1. there are really two fingers which occasionally merely look like one (the rationalists, and where the one, so-called, is an illusion).

2. there is something which sometimes is one and sometimes splits into two and which, as far as we know, might split into three or twenty. (the empiricist, and where there is no illusion.)

3. there is a single object which merely looks like two due to the amalgamation of two points of view in the brain (the common sense position [Kant] and where the only illusion is to think, like the empiricist: that there is no illusion; and like the rationalist: that the single finger is an illusion and the two fingers reality).

With regard to a shadow, these same people would say, respectively:

1. there is really a shadow all the time, only you cannot see it in the dark due to the lack of light (the rationalist, and where it is an illusion to think there is no shadow).

2. shadows are things which appear and disappear on their own, and as far as anyone knows might suddenly appear in bright red or dripping wet at the utter lack of any light at all (the empiricist position [and especially as delineated by Berkeley]).

3. the shadow is always opposite some object from the brightest spot around, and is a lack of something, i.e., it is a negative, the lack of these things called light waves which emanate from the brightest spot (called the light source) and travel to the object and then bounce back off of that object into the eye, such that when an object stands between the light source and the lighted surface, e.g., a wall, there is an interruption of the light waves in precise proportion to the object, and the resultant negation takes on the shape of the interrupting object (call the object casting the shadow). What we call the dark then is nothing other than this same effect, i.e., the light source is hindered from reaching the reflective surface (hindered by not being on at all).*

[* Which, to hear it said, seems far more fantastic and contrived than the stories of either the rationalist or the empiricist.]

Kant's Choice of 7+5 = 12 for his Example of the Need for a Pure Envisagement for Mathematics.

This is a matter of some interest for my speculation. I am assuming that Kant uses all words and all (of his agonizingly few) examples very deliberately. Thus I assume that there is a lesson in the example of 7+5=12.

Now while several things come to mind, e.g., 12 takes us beyond the ten fingers of our hand, by far the most important, in my opinion, is the fact that the 12 is the lowest number which could be obtained definitively without having been given in advance. I shall try now to make this point clear.

The enumeration rule must give me the elements of the numbering system, e.g., 1 through 9. But if this is all that I had, I could not possibly know what followed the 9 in counting. It might be a & or a @ or anything. Thus when I learn that it is a 10, then this is news to me. And it is likewise news to me that an 11 follows the 10 *. But it is not news that 12 then follows 11, for the numerals of 1 through 11 give me the pattern (assuming that the enumeration does follow a pattern)** For one of the most central components of the human mind is the recognition of patterns (which are always given in the envisagement/sighting). Thus if I wanted to give the gist of our (Arabic) numeration system briefly I would merely give the first 11 counting numerals, for no more is necessary. I.e., if another being cannot catch on with those, heshe cannot catch on with any.

[* For it could have been a 20, for all I know, i.e., 9 10 20 30 ... 90 100 200, etc.]

[** This is a basic assumption for the practical validation of mathematics, I suppose. Assume, for example that the numbers went so ... 941, 942, &&, 943, ... In that case we would never know what were coming next and would never be sure of the consistency of mathematics except that all elements be explicitly defined, e.g., &&=942+1 and 943=&&+1, etc.]

I have no similar explanation for the use of the 7 and the 5. I assume that Kant would not want to utilize 2 sixes due to the possibility of a confusion with regard to which number (the 7) is held mentally and which (the 5) is depicted by the hand. Perhaps he wanted the all the remaining digits (the 5) to fit on one hand so that the other hand could be used to point with. Or perhaps (and here I am beginning to grope) it is only in the picture of all outstretched fingers that the emphasis can be placed on the envisagement and sighting of the five. For with all five fingers outstretched, it would be possible to see instead merely a hand, while if only 4 or less were shown, it would seem more intuitive. And it is my opinion that the German Anschauung (envisagement) suggests a dual way of looking at the same object, in this case with a hand or a five,* and this equivocation would be most pronounced with a five presented in space than with any other number, and thereby suggest the notion of the envisagement more clearly to the student of Kant's philosophy.

[* It is instructive, I think, for students of Kant to consider that in looking at this picture of the outstretch fingers that it is possible not to see the representation of a 5 at all, but rather long (or stubby) fingers, or dirty finger nails or the front (or back) of a hand, or a handsome ring, or discolored skin, etc., etc., (and perhaps etcetera without limit). This is certainly part of the suggestion of the envisagement/Anschauung, that what you see is a function (partially, at least) of what you are looking for and what you have in mind.

The Formulation of Arithmetic as a Logical Science

With arithmetic we are dealing with with definitions and with rules of manipulation.

I. Definitions:

1. Expressions are of two sorts, either they are

A. names (and are also called numerals or simple expressions),

and are either

a. counting numerals, i.e., 1 2 3 4 5 6 7 8 9 10 11, etc. or

b. 0;

or else they are

B. nicknames (or composite expressions) which consist of two or more numerals, each of which will be preceded by a combinatory mode or operand (although the operand preceding the first of which may be implicit). [For example: Mac may be the nickname of Philip McPherson.]

The first nine counting numerals along with 0 are called the fundamental elements of the science..

2. The operands are + and -

Thus 10 is a name and 7+5 is a nickname, as is 13-4. Whether a nick name is a nickname of a given name can be determined via the rules of composite expression (see below). It is impossible to tell via intuition that 7+5 is or is not a nickname of 11, for example. If 7+5 = 11, then 7+5 is said to be a nickname of 11.

Any name and the expressions which are nicknames of that name are called synonyms of each other.

3. Numerals following a given numeral in the enumeration rule (I.1.A.a. above) are greater than that given numeral, and numerals preceding a given numeral are less than that numeral.

4. 0 immediately precedes 1.

5. 1 immediately follows 0.

II. Rules

1. The Plus Operand

A. If the first numeral of any expression is prefixed by a +, the + may be removed

B. If the first numeral of any expression has no prefixed operand, a + may be added.

Thus, for example, +8-5 = 8-5, and 8 = +8

2. Rearrangement of Composite Expressions.

A. All numerals of a composite expression which are prefixed by an operand may, together with their respective operands, be rearranged as desired.

For example -7+8 = +8-7

3. The Zero

A. Any numeral consisting entirely of one or more zeros may be removed along with any prefixed operand.

Thus -0 may be eliminated so that 8-0 = 8.

B. A zero may be added to any expression and prefixed with a + or a -.

Hence 8 = 8-0 = 8+0

4. Substitutions

A. Every numeral may be replaced by the numeral immediately preceding it (according to I.1.A.a., and I.4) along with a +1.

Thus 4 = 3+1

B. Any numeral and a +1 may be replaced together by the next numeral (according to I.1.A.a, and I.5)

Thus 3+1 = 4

Application to 7+5

7+5 = 7+4+1 (per II.4.A) = 7+1+4 (per II.2.A) = 8+4 (per II.4.B) = . . . = 12

Conclusion

And thus can we see that 7+5 and 12 are synonyms, although this was only possible by virtue of a peek at the object, i.e., the numerals arrayed in space; for the 12 was not defined; and the look at the numerals only revealed the pattern of the numerals, and no other information was gleaned from the look.

It is also revealing to consider that the substitutional rules above, namely: II.4.A and II.4.B, are axioms, and their validity is not apparent except by a sighting of the object, namely the counting numerals arrayed along a number line. For these substitution rules are really nothing more than rules for making definitions, and that these definitions will not conflict cannot be ascertained from the rules, but only from an inspection of the rules as sighted in the form of the pattern of the counting numbers and then as arrayed on a line.

Essentially then it is not possible to recognize logically that 7+1 and 1+7 are substitutes;* and thus an inspection is required, albeit an inspection via a pure envisagement (reine Anschauung) which is called space.

[* For otherwise we would know logically that a chair and a brown table would be the same as a table and a brown chair, and that may not be true at all! And so it certainly is not true merley by virtue of the logic, but only via a(n empirical) sighting. Kant's favorite proof of this fact is given in the concept of incongruent counterparts and is presented in the following appendix regarding conversations with a blind person about matters spatial.]

Such an axiom is no different than any axiom of geometry where, for example, we say that two parallel lines can be joined at any point on each with a line perpendicular to each, and any two such lines are equal in length. And this parallel axiom, in turn, is the basis for the assertion that objects do not diminish in size as we increase our distance from them.*

[* That this is an axiom is clear from the fact that the impression made is that objects decrease in size as the distance between them and the viewer increases, at a diminishing rate of decrease, even as, for example, the objects (as specters) nearest a rapidly moving automobile move in the opposite direction from the automobile very rapidly, while those more distance move more slowly and finally very distant objects, the moon, don't move at all (or, shall we say, move in the direction of the automobile).]

Conversation with a Blind Friend

The look of things is more powerful than the logic of things, really, and this look of things is the basis of mathematics. To suggest the truth of this assertion I shall speak to a perfectly rational man who is blind and indeed from birth, i.e., has no memory of any visual and spatial images. I begin by conceiving of a particular object in space, namely two triangles on the surface of a sphere, which share a common base, AB, and the other two sides of the one, i.e., BC and CA, are unequal, but they are equal, respectively, to the two sides of the other, i.e., BC = BC' and CA = C'A. [While not critical it is often helpful to let C and C' be on opposite sides of AB so that the image is that of two, equal, spherical triangles in a juxtaposition called reflection, like that of my left hand and the mirror image of my left hand.*]

[* It is important for the reader to keep in mind that no two sides of either of the triangles are equal, i.e., AB <> BC, AB <> CA, and BC <> CA. Otherwise the logic of the conversation will not hold. For example if the triangles were isosceles or equilateral, they will be found to be substitutable.]

Now I note first that each of the sides of each triangle can be substituted for its counter part in the other triangle, and without the least difference, perceptible or otherwise, i.e., BC for BC' and CA for C'A.

Now I begin my conversation with my blind friend.

"Look," I say, "I have two objects and I want to know whether I can substitute one for the other, like I can substitute the names, Philip and Phil, and not make any difference. If it does not make any difference then one is a substitute of the other.

"I note that ABC consists only of AB, BC and CA, nothing more. These three elements are the parts of the ABC and comprise it totally. Likewise ABC' is composed entirely and only of AB, BC' and C'A.

"I note further that AB is one and the same thing. And that BC and BC' can be substituted for each other and it would be absolutely impossible to tell that such a substitution had taken place. And the same holds true of CA and C'A.

"Now given that, given that every part of one can be substituted for every part of the other and that it would be absolutely impossible to tell that this substitution had taken place, can we conclude that the two triangles can be substituted, one for the other?

"In other words, if we substitute all of the parts of one for all of the parts of the other, have we substituted the wholes?"

And my blind friend will answer, "Why yes, of course! obviously!, for what else is the triangle (according to what you told me) except the sum of its parts, and if all of the parts have been substituted, then of course the whole has been substituted."

And I reply, "But that is not true, for I cannot place the one triangle in the place of the other, no matter how hard I try."

And my blind friend replies, "But that doesn't make any sense."

And I reply in turn, "I know; but if you could see it, you would see that it is true and that it does make sense. But you do have to see it!"