Plato’s Divided Line Analogy
Michael D. Capistran
Presented to the Raleigh Tavern Society November 29, 2001
I wish to thank you for the opportunity to give this paper. This is a paper I have been intending to write for a while. In reviewing the material, however, I discovered recently that the fundamental idea I had in mind has already been proposed by my own favorite scholar on Plato, Cornford.[1] It is therefore surprising Cornford’s suggestion has not been taken seriously by other scholars. As but one example of this general misunderstanding, Richard Robinson looses himself in the conundrum how it is possible Plato’s hypothetical method may achieve absolute knowledge:
…when we recall that he [Plato] believed in the possibility of absolute knowledge, we may think that he ought to have regarded it [dialectic]as a tenth best, and we must certainly wonder why he devoted so much space to its elaboration of a method for winning the real and not impossible prize, infallible certainty.”[2]
I’m afraid I shall be doing little more than adding elucidatory, updated examples to illustrate Cornford’s explanation.
The question I shall be addressing is rather a traditional puzzle in the history of thought. Why, in the Divided Line passage of the Republic, does Plato relegate mathematics (geometry and arithmetic) to dianoia rather than noēsis? That is, why is mathematics considered a method of the lower Forms rather than a method of the higher Forms, and why is mathematics, a discipline we know with certainty, grouped with a lower kind of understanding than that of Socratic dialectic, a kind of thinking that never seems to lead to any conclusions.
Plato’s Theory of Forms
Here, briefly, is Plato’s Theory of Forms.
Plato believed in two realms of reality, a higher and a lower. He believed the objects in the lower realm were physical objects and the objects in the higher realm were “Forms” or “Eidos.” He believed the Forms were perfect and were more real than the physical objects. He thought that whereas physical objects were spatio-temporal, changing and changeable, sensible, and copies of Forms, Forms were transcendent, eternal, intelligible, and archetypal. Plato thought that the relation that held between physical objects and Forms was that physical objects “imitated” and “participated in” Forms.
To a person first encountering this theory, the notion of a higher realm with objects impossible, even in principle, to be perceived by the senses will undoubtedly seem strange.
As a heuristic or explanatory example, consider the circle. The top of a glass tumbler and the face of a watch are both circles. Here is another circle:
This is a physical circle appearing in time and space, on a physical piece of paper. The line from the center of the circle to its circumference is referred to as the radius, thus:
Geometric figures such as circles have properties. As the radius of a circle increases, for example, the circle itself expands; as the radius decreases, the circle contracts. That is to say, it is a property of the circle, a theorem, that there is a direct proportional ratio between the radius of the circle and its circumference.
Geometry is a discipline that has its practical application in any culture. In ancient Egypt, for example, the Nile would overflow its banks each year and the land would have to be surveyed for reapportionment. Though geometry also had its practical application for the ancient Greeks, the Greeks took enormous interest in a research program, spanning generations and even centuries, to discover the construction and properties of the basic geometric figures.
The ancient Greek geometers would discover such properties with the help of drawing figures in sand as we would on a blackboard or on a piece of paper. But if a property such as that given above (i.e., there is a direct proportional ratio between the radius of the circle and its circumference) holds – is true – what is it true of? Is it true only of the circle drawn in the sand, on the board, or on the paper – or the face of a watch, or the top of a glass tumbler? The ancient Greek geometers thought not. They believed that these circles were but the physical and imperfect manifestations of an abstract, ideal, geometric figure, an object of pure logic – the circle itself. These geometers believed they were using the drawings in the sand – physical drawings within the context of time and space – to help them reason about the ideal properties of the ideal figures.
Again, since the ancient Greeks were making discoveries, it seemed natural for them to believe they were making discoveries of something. That is, if we have discovered a property, is it not reasonable to believe this property we have discovered is a property of something? If the activity of running is occurring, is it not reasonable to believe there is a runner?
Consider this. The physical circle comes into existence, goes out of existence, and is subject to change. The eternal circle, should we grant its existence, never changes. If every single physical circle were eradicated from the physical realm, would the property of the circle we mentioned above not still be true? Yes. It would still be true. Was it true yesterday? Yes. Will it be true tomorrow? Yes. When was it not true? It’s a property that is always true. It is eternally true. If it is eternally true, what is it true of, if not an eternal thing – an eternal circle? Which of the two circles, the physical or eternal and ideal, is more real? Being eternal and unchanging, independent of the physical, one might, and Plato did, believe the ideal circle is more real than the physical circle.
Plato also took this opinion concerning geometric figures and universalized it. For Plato, not only do mathematical entities such as circles have metaphysical status, but other abstract entities exist as ideals as well.
As an example, what is it that you and I have in common that makes us the same kind, type, or species of thing? This is the fundamental question a theory of Forms is designed to answer: what is it that differentiates kinds or types? Plato suggests that the reason you and I are the same type of thing is that we share an essence or structure. We are both more or less imperfect copies of the same ideal blueprint – the structure of human being – in the same way that the physical circle emulates the ideal circle.
The same also holds for virtues. For Plato, ethical standards are as absolute as geometric standards. I can throw down my weapons in battle and run away, and I can even do so and call, or name, it courage. What I cannot do is make a cowardly act into a courageous one. The virtues such as piety, temperance, courage and justice may be difficult to come to know, but this does not keep them from being abstractly eternal, ideal, and unchangeable.
The Divided Line Passage from Book VI of Plato’s Republic
Here is the celebrated passage I wish to examine. Socrates is having a discussion with Glaucon, Plato’s brother.
[Socrates
explains to Glaucon:] You surely apprehend the two types [of entities] the
visible and the intelligible.
I
do.
Represent
them then, as it were, by a line divided into two unequal sections and cut each
section again in the same ratio – the section, that is, of the visible and that
of the intelligible order – and then as an expression of the ratio of their
comparative clearness and obscurity you will have, as one of the sections of
the visible world, images. By images I
mean, first, shadows, and then reflections in water and on surfaces of dense,
smooth, and bright texture, and everything of that kind, if you apprehend.
I
do.
As
the second section assume that of which this is a likeness or an image, that
is, the animals about us and all plants and the whole class of objects made by
man.
I
so assume it, he said.
Would
you be willing to say, said I, that the division in respect of reality and truth
or the opposite is expressed by the proportion – as is the opinable to the
knowable so is the likeness to that of which it is a likeness?
I
certainly would.
Consider
then again the way in which we are to make the division of the intelligible
section.
In
what way?
By
the distinction that there is one section of it which the soul is compelled to
investigate by treating as images the things imitated in the former division,
and by means of assumptions from which it proceeds not up to a first principle
but down to a conclusion, while there is another section in which it advances
from its assumption to a beginning or principle that transcends assumption, and
in which it makes no use of the images employed by the other section, relying
on ideas only and progressing systematically through ideas.
I
don’t fully understand what you mean by this, he said.
Well,
I will try again, said I, for you will better understand after this
preamble. For I think you are aware
that students of geometry and reckoning and such subjects first postulate the
odd and the even and the various figures and three kinds of angles and other
things akin to these in each branch of science, regard them as known, and,
treating them as absolute assumptions, do not deign to render any further account
of them to themselves or others, taking it for granted that they are obvious to
everybody. They take their start from
these, and pursuing the inquiry from this point on consistently, conclude with
that for the investigation of which they set out.
Certainly,
he said, I know that.
And
do you not also know that they further make use of the visible forms and talk
about them, though they are not thinking of them but of those things of which
they are a likeness, pursuing their inquiry for the sake of the square as such
and the diagonal as such, and not for the sake of the image of it which they
draw? And so in all cases. The very things which they mold and draw,
which have shadows and images of themselves in water, these things they treat
in their turn as only images, but what they really seek is to get sight of
those realities which can be seen only by the mind.
True,
he said.
This
then is the class that I described as intelligible, it is true, but with the
reservation first that the soul is compelled to employ assumptions in the
investigation of it, not proceeding to a first principle because of its
inability to extricate itself from and rise above its assumptions, and second,
that it uses as images or likenesses the very objects that are themselves
copied and adumbrated by the class below them, and that in comparison with
these latter are esteemed as clear and held in honor.
I
understand, said he, that you are speaking of what falls under geometry and the
kindred arts.
Understand
then, said I, that by the other section of the intelligible I mean that which
the reason itself lays hold of by the power of dialectic, treating its
assumptions not as absolute beginnings but literally as hypotheses,
underpinnings, footings, and springboards so to speak, to enable it to rise to
that which requires no assumption and is the starting point of all, and after
attaining to that again taking hold of the first dependencies from it, so to
proceed downward to the conclusion, making no use whatever of any object of
sense but only of pure ideas moving on through ideas to ideas and ending with
ideas.
I
understand, he said, not fully, for it is no slight task that you appear to
have in mind, but I do understand that you mean to distinguish the aspect of
reality and the intelligible, which is contemplated by the power of dialectic,
as something truer and more exact than the object of the so-called arts and
sciences whose assumptions are arbitrary starting points. And though it is true that those who
contemplate them are compelled to use their understanding and not their senses,
yet because they do not go back to the beginning in the study of them but start
from assumptions you do not think they possess true intelligence about them
although the things themselves are ineligibles when apprehended in conjunction
with the first principle. And I think
you call the mental habit of geometers and their like mind or understanding and
not reason because you regard understanding as something intermediate between
opinion and reason.
Your
interpretation is quite sufficient, I said.
And now, answering to these four sections, assume these four affections
occurring in the soul – intellection or reason for the highest, understanding
for the second, belief for the third, and for the last, picture thinking or
conjecture – and arrange them in a proportion, considering that they
participate in clearness and precision in the same degree as their objects
partake of truth and reality.
I
understand, he said. I concur and
arrange them as you bid.[3]
Reason (noesis) Higher Forms
Knowledge Intelligible
Understanding (dianoia) Lower Forms
Belief (pistis) Physical Objects
Opinion 
Visible
Picture Thinking (eikasia) Images
Deduction, Induction, and
Abduction
Peirce, the greatest American philosopher, suggests there are three types of reasoning: Deduction, Induction, and Abduction or Hypothesis. Deduction is the inference of a result from a rule and a case.
Rule. – All the beans from this bag are white.
Case. – These beans are from this bag.
¬ Result. – These beans are white.
Induction is the inference from the rule from the case and result.
Case. – These beans are from this bag.
Result. – These beans are white.
¬ Rule. – All the beans from this bag are white.
Abduction is the inference of a case from a rule and a result.
Rule. – All the beans from this bag are white.
Result. – These beans are white.
¬ Case. – These beans are from this bag.
This third type of reasoning might require a bit more elaboration. Here is what Peirce says:
Suppose I enter a room and there find a number of bags, containing different kinds of beans. On the table there is a handful of white beans; and, after some searching, I find one of the bags contains white beans only. I at once infer as a probability, or as a fair guess, that this handful was taken out of that bag. This sort of inference is called making a hypothesis.[4]
Deduction
Deduction is an inference in which the conclusion must follow from the premises as a matter of necessity, or, more technically, an inference in which the premises are claimed to support the conclusion in such a way that it is impossible for the premises to be true and the conclusion false.
Deduction is, in other words, reasoning rigorously from principles. We use deduction to prove certain things must necessarily be a certain way. We use deduction, for example, to prove theorems in geometry. I have chosen as an example of deduction Euclid’s first use of the reductio ad absurdum. Before I give the theorem, however, we should first review the principles upon which the theorem is based. I have truncated the definitions of Book I as we shall not need them all:
Book I
Definitions.
1. A point is that which has no part.
2. A line is breadthless length.
3. The extremities of a line are points.
4. A straight line is a line which lies evenly with the points itself.
5. A surface is that which has length and breadth only.
6. The extremities of a surface are lines.
7. A plane Surface is a surface which lies evenly with the straight lines on itself.
13. A boundary is that which is an extremity of anything.
14. A figure is that which is contained by any boundary or boundaries.
19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.
20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.[5]
Postulates.
Let the following be postulated:
1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and distance.
4. That all right angles are equal to one another.
5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Common Notions
1. Things which are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.
Here is our theorem, Proposition 6 of Book I:
If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another.
A
D
B C
Let ABC be a triangle having the angle ABC equal to the angle ACB;
I say that the side AB is also equal to the side AC.
For, if AB is unequal to AC, one of them is greater.
Let AB be greater; and from AB the greater let DB be cut off equal to AC the less;
Let DC be joined.
Then, since DB is equal to AC, and BC is common,
the two sides DB, BC are equal to the two sides AC, CB respectively;
and the angle DBC is equal to the angle ACB;
therefore the base DC is equal to the base AB,
and the triangle DBC will be equal to the triangle ACB,
the less to the greater:
which is absurd.
Therefore AB is not unequal to AC;
it is therefore equal to it.
Therefore etc.
Q.E.D.
Our conclusion is drawn with absolute certainty – but is nevertheless entirely dependent upon the axioms (definitions, postulates, and common notions) from which we start. The Greeks were aware that the type of system one ended up with in geometry was dependent upon the definitions and postulates one started with. Plato was familiar with this, as was Aristotle.[6] Let me use an anachronistic example to help us understand this point. What is referred to as the “parallel postulate” has always been problematic. Do parallel lines meet or do they not? In the Nineteenth Century this problem was investigated with a high degree of interest and rigor. There are two methods of proof in logic: the direct and the indirect proof. The example given above is an example of indirect proof. With the indirect proof, we assume the opposite of what we are trying to prove, find a contradiction, and thereby show our assumption – the opposite of what we are trying to prove – cannot be correct. If there are only two options, this then demonstrates that what we are trying to prove is correct. You will notice the similarity between Platonic dialectic and this method: we assume and then search for a contradiction. It has always been known to geometers that if the parallel postulate is unsound, by assuming its opposite – essentially that parallel lines do meet – one should be able to come up with a contradiction. Reimman tried this and found, to the contrary, no contradiction occurs. Rather what happens is we have an entirely different and coherent system with different rules. In fact it was discovered that there is an entire spectrum of spaces or systems and the system of Euclid – in which parallel lines do not meet – is only a central boundary of these systems. The ancient Greeks were not, of course, familiar with non-Euclidean geometries or spaces, though they were very curious concerning the status of the parallel-line postulate. I have only used this example as a heuristic and exaggerated one to help us understand that the system we end up with is dependent upon the postulates and definitions with which we start.
I’m sorry to have dragged you through this, but I believe it is easy now to understand what Plato was talking about when he said that dianoia was the type of thinking that advances “by means of assumptions from which it [the soul] proceeds not up to a first principle but down to a conclusion.”
For I think you are aware that students of geometry and reckoning and such subjects first postulate … the various figures and three kinds of angles and other things akin to these …, regard them as known, and, treating them as absolute assumptions, do not deign to render any further account of them to themselves or others, taking it for granted that they are obvious to everybody. They take their start from these, and pursuing the inquiry from this point on consistently, conclude with that for the investigation of which they set out.
Notice that we not uncommonly draw a figure or picture, a physical drawing residing in time and space, to help us understand the abstract and ideal logic of the proposition in question. This is what Plato means when he says:
And do you not also know that they [the students of geometry] further make use of the visible forms [visible, physical figures which are drawn] and talk about them, though they are not thinking of them but of those things of which they are a likeness [the ideal, abstract, geometric figures], pursuing their inquiry for the sake of the square as such and the diagonal as such, and not for the sake of the image of it which they draw? And so in all cases. The very things which they mold and draw, which have shadows and images of themselves in water, these things they treat in their turn as only images [of Forms], but what they really seek is to get sight of those realities which can be seen only by the mind.
Thus, certain of the Forms are considered of the lower rather than the higher sort for two reasons: 1. They are deduced from preconceived assumptions, and 2. We (often) use physical pictures to help in our understanding of them:
This then is the class that I described as intelligible, it is true, but with the reservation first that the soul is compelled to employ assumptions in the investigation of it, not proceeding to a first principle because of its inability to extricate itself from and rise above its assumptions, and second, that it uses as images or likenesses the very objects that are themselves copied and adumbrated by the class below them, and that in comparison with these latter [physical drawings] are esteemed as clear and held in honor.
How we come up with the principles or assumptions themselves, with which we start, however, is a different matter. This leads us to abduction.
Abduction
Consider, for a moment, the evidence available to Watson and Crick before their discovery of the structure of the DNA molecule:
They knew the DNA could form crystals, and so must have a repeating, regular, orderly structure; that is, they knew the structure could be solved.
They knew Pauling’s rules, the basic grammar of molecular combination.
They knew the composition and structure of the component molecules: ribose, phosphate, and nucleotide. For example, they knew the chemical structure of the sugar-phosphate-base: third carbon in the sugar ring linked to the phosphate and that in turn linked to the fifth carbon of the next sugar, and so on.
They knew the Van der Walls distances, the boundary limits of the contacts the chemicals must have with other molecules.
They had the alpha and beta X-ray crystalographs, photographs of the unstretched and stretched (wet) structures. These, within the context of projective geometry, defined the physical limits the structure would have to meet or be compatible with.
The alpha X-ray diffraction photograph provided three limiting measurements: the width of the structure, the height of each specific iteration (the stacked, parallel bases), and the height of general iteration (one complete turn of the structure). That is, it showed a 3.4 angstrom spacing between nucleotides and a larger structural repeat of some sort at 27 angstroms.
The beta X-ray diffraction photograph expressed forcefully a helical structure. It also told them there was a full iteration at 34 angstroms, after, therefore, exactly ten sets of nucleotides, and that the diameter of the helix was 20 angstroms.
They were familiar with the Chargaff ratios, that total purines to pyrimidines, and also adenine to thiamine and guanine to cytosine, were at ratios of 1 to 1.
They also knew the structure has to be monoclinic, face-centered – that is, the chains had to be running in opposite directions, which in turn meant the chains had to be dual or at least even in number.
They suspected the Furberg conjecture, that the sugar would be parallel to the nucleotide base, was correct.
Optimally, also, the structure should not be “dull”; it should be both auto- and hetero-catalytic.
Anyone wishing to come up with the structure had to produce a hypothetical conjecture, a model, that would be in conformity with all or most of this data. The notion of abduction is captured by Linus Pauling when he said, concerning the alpha helix, “I asked what the structure should be in order to account for the way it takes up oxygen.”[7]
There were five primary attempts to postulate the structure of the DNA molecule. Each attempt, it turned out, resulted in a progress in sophistication. Each attempt got closer to the actual structure.
In 1947, Astbury published the best X-ray diffraction pattern to date for DNA, together with a general proposal of its molecular structure. He suggested that both the sugar components and nucleotides were stacked in a non-spiral fashion. Clearly not in conformity with much of the data, the suggested structure did emphasize a 3.4-angstrom spacing between nucleotides and a general repeat at 27 angstroms.
In 1949, Furberg, on the basis of his own X-ray studies, reacted to Astbury, setting sugar perpendicular to nucleotide base. “Furberg’s nucleotide – correcting Astbury’s error – was absolutely essential to us,” has said Crick.[8] Furberg’s best proposed structure, a single chain in helical form with the bases sticking out flat and parallel to each other, retained Astbury’s correct distances, but was insufficiently dense. There was too little in it.
Not long after Watson and Crick first met, they proposed a structure, but Watson at a seminar had misheard, or misremembered, the amount of water required and they had therefore gotten the structure wrong. This mistake, though embarrassing, helped considerably to focus their attention upon the problem.
Some weeks before Watson and Crick’s final suggestion, Pauling and Corey published a proposed structure for DNA. Watson and Crick recognized immediately the proposed structure could not be right. Pauling and Corey had the wrong value for the density, and their suggestion was not in conformity with the Franklin X-Ray diffraction photographs.
I must pause here. This process or situation is entirely analogous with – in fact the same procedure as – dialectic as proposed by Plato.
We have many examples of Plato’s method of dialectic. In seeking the nature or defining characteristic of a specific virtue we first postulate, propose, or suggest a definition of that virtue. The definition is then subjected to scrutiny and critical analysis. If a contradiction is found, the definition is thrown out and another, different, definition is proposed. The new definition is better than the previous one in the sense that, not only has it not incorporated the same mistake as the original, but a recognition of some of the problems associated with formulating the definition helps us understand a bit more about the nature of the virtue in question. The same procedure is followed with the new definition, once again testing for contradictions. If a contradiction is found, the definition is again tossed out and another definition is postulated, and so forth. We thus progress to the essence or true nature – the true definition – of the virtue in question. We conjecture on the basis of that with which we feel ourselves to be familiar. As an example, in Laches Laches and Socrates both have the attribute of courage and are therefore in a position to propose a suggestion for the logos, or structure of courage. In each instance within the Socratic dialogues the consequences of the suggested answer or definition, when followed out, lead to a contradiction. Each definition is therefore recognized as deficient, and is thrown out. Pauling and Corey’s conjecture contradicted the “facts” of known density and diffraction. If the conjecture results in a contradiction, we throw the conjecture out and start again, but this time with the additional knowledge of how the conjecture went wrong.
This procedure may sound novel but is actually the procedure by means of which any discipline, including philosophy, progresses. Peer review in scholarship follows, optimally at least, this method. It is not uncommon, for example, for philosophers to object to this procedure, but such objections, being theoretical in character, follow exactly this procedure in formulating the objection.
Plato’s method of dialectic includes criticism, modification, and the development of consequences. We find out what is right, not by deducing it right away, but by making a conjecture or hypothesis and then going about trying to discover why the conjecture can’t be right. As dialectic in this sense is identical with abduction – the projection and testing of a hypothesis – we would be hard-pressed to say dialectic was not operative in the discovery of the structure, definition, the logos of the DNA molecule.
In the words of Crick:
…from Bragg and Pauling I learned how to see problems, how not to be confused by the details, and that is a sort of boldness; and how to make oversimple hypotheses – you have to, you see, it’s the only way you can proceed – and how to test them, and how to discard them without getting too enamored of them. All that is a sort of boldness. Just as important as having ideas is getting rid of them.[9]
The structure, by the way, originally suggested by Watson and Crick has itself received further elaboration; that is, the structure as first accepted by the scientific community is not quite that accepted today.
A similar situation held within the context of ancient Greek geometry. We can easily imagine the ancient Greeks following this process in the discovery of the definitions, postulates, and common notions of Euclidean Geometry. This project would have been still an ongoing one during Plato’s lifetime. We are familiar there were a number of different versions of the Elements of Geometry in ancient Greece, but the only one that has come down to us Euclid’s – of which we have multiple copies. Euclid was the famous synthesizer whose work was so masterful that all the others were thrown out. Though the ancient Greeks had been doing this kind of geometry since Thales (580 B.C.E.), Euclid (fl. 300 B.C.E.)does not predate, but post-dates, Plato (d. 347 B.C.E.). Euclid comes, in fact, after Aristotle. We are aware that there were differences of opinion , not in the overall nature or approach of the discipline, but on the exact nature or expression of the basic principles or starting points. For example, as I have mentioned, the parallel-line postulate was thoroughly debated and re-debated. Again, it is easy to imagine the various definitions, postulates and common notions being themselves hammered out through the method Plato refers to as dialectic –as it were by an enormous, multi-generational committee of scholars.
In terms of discovery, without abduction we have a tendency to thrash around blindly. An example of this would, I take it, be when Bragg, Kendrew, and Perutz published a list of nearly twenty possible polypeptide bonds, then tried to choose the most likely one – not the structure that it must be. They got the structure wrong, and shortly afterwards Pauling published the correct structure.[10]
This is not to say that such procedure might not produce results. Archimedes is said to have immersed metal figures in water to find which theorems he should seek to prove. Nevertheless, such thinking is not optimal thinking. Setting up a lot of random possibilities and selecting the “best” does not allow for the kind of “thinking outside the box” that is abduction. The situation here seems to me quite similar to the attempts on the part of cognitive scientists to prove a computer is capable of creative thought because it can quickly sift through many possibilities.
Substantiation
The substantiation of Watson and Crick’s proposed structure came from an experiment in density-gradient centrifugation. In a solution, a soup, of heavy nitrogen, N15, Meselson and Stahl grew E. coli for fourteen generations at blood heat. The heavy nitrogen was therefore built into the bases of all the DNA. Half of the bacteria were removed and placed in a solution of normal nitrogen, N14. A control of bacteria fed only normal nitrogen was also grown.
Now, several times during each of several more generations, they took another sample. Finally, as a reference point, they made up a separate sample of ordinary, unlabelled DNA. Each sample was quickly chilled, centrifuged lightly for five minutes to drive the bacteria into a pellet. The cells were then broken up, by adding a detergent, to release their DNA; a measured portion of each sample was put into cesium chloride solution, and each tube spun at 44,770 revolutions per minute for twenty hours.
The DNA from the first sample, harvested before the lighter nitrogen was introduced, formed a crisply defined band towards the denser, lower end of the gradient. The DNA from the unlabelled reference sample formed a band slightly higher up. When the two types were mixed and spun, they separated; two bands formed with a space between.[11]
That is, the results were one band for bacteria grown on heavy nitrogen only; one band for bacteria grown on normal nitrogen only; or two bands for a combination of some bacteria grown on heavy and some bacteria grown on normal nitrogen.
In the following generations, if DNA replication were dispersive (came entirely apart for replication) there would only be one band in the middle position of average density. The N15 and N14 would combine without differentiation. If DNA replication were conservative (nothing altered in the chemical composition of the original molecule), there would always be two bands, the labeled old molecules and the unlabelled new molecules, and never a band of mixed density. Semiconservative replication, however, predicts a medium level gradation in first generation as the normal nitrogen is introduced. That is, the original half-strand remains and a new half-strand is added. The second generation should also have no heavy nitrogen strands, fewer hybrid strands, and a new, higher, level of strands constructed only with normal nitrogen. As reported by Meselson and Stahl:
The degree of labeling of a partially labeled species of DNA may be determined directly from the relative position of its band between the band of fully labeled DNA and the band of unlabeled DNA…. It may be seen … that, until one generation time has elapsed, half-labeled molecules accumulate, while fully-labeled DNA is depleted. One generation time after the addition of N14, these half-labeled or “hybrid” molecules alone are observed. Subsequently, only half-labeled DNA and completely unlabeled DNA are found. When two generation times have elapsed…, half-labeled and unlabeled DNA are present in equal amounts.
The nitrogen of a DNA molecule is divided equally between two subunits which remain intact through many generations…. Following replication, each daughter molecule has received one parental subunit…. The results of the present experiment are in exact accord with the expectations of the Watson-Crick model for DNA duplication.[12]
The process of substantiation works as follows. The proposed theory, the hypothesis, makes certain predictions. We deduce these consequences and test for them. If the consequences are borne out, the hypothesis receives a modicum of substantiation. The higher the specificity of the proposed theory, the higher the degree of subsequent substantiation.
In other words we reason upwards by using hypotheses as springboards to reach our ultimate conclusion – after the fashion of the multiple attempts to solve the DNA structure. This conclusion is reached by proceeding from one theory, one thought, to another theory, another thought. When we have our final proposal, Watson and Crick’s conjecture in this case, we reason downwards once again, using our thoughts to predict and test for subsequent consequences. Plato:
Understand then, said I, that by the other section of the intelligible I mean that which the reason itself lays hold of by the power of dialectic, treating its assumptions not as absolute beginnings but literally as hypotheses, underpinnings, footings, and springboards so to speak, to enable it to rise to that which requires no assumption and is the starting point of all, and after attaining to that again taking hold of the first dependencies from it, so to proceed downward to the conclusion, making no use whatever of any object of sense but only of pure ideas moving on through ideas to ideas and ending with ideas.
A further example of substantiation – the deduction of specific results or effects from hypothetically postulated principles – would be Max Perutz’s appreciation that past X-ray diffraction pictures had employed too small a photographic plate, and the wrong angle between the fiber and the X-ray beam, to find a smudge at the spot of a 5.44 angstrom turn. On the afternoon after the morning Perutz read Pauling’s paper on the alpha helix, he placed a horse hair down the center of a cylindrical sheet of film at a calculated angle and found the predicted spot – at 5.44 angstroms, not 5.1. The same spot was also subsequently discovered in other polypeptides.[13]
This leads us then to induction.
Induction
This is not a paper on the two roads to Larissa:[14] knowledge and true opinion. I have argued that Plato’s two types of knowledge are deduction and abduction. The suggestion that belief or faith (pistis) conforms to induction sits well with what Socrates has said. Generalizations concerning physical objects drawn by means of the senses may yield only phenomenal laws and phenomenal laws may never be more than conjectures. Ideal laws are what science seeks.
During the discovery process, Crick was highly cognizant that DNA as dried and photographed might actually turn out to be nothing like wet, active DNA at all. The X-ray patterns, for example, might just turn out to be an artifact of the preparation methods. Specifically, any of the above listed “facts” might, he was very aware, turn out to be entirely incorrect after all. For this reason, Crick made a conscious effort to try to think of the potential structure in terms of the fewest possible parameters. Crick:
You must remember, we were trying to solve it with the fewest possible assumptions. There’s a perfectly sound reason – it isn’t just a matter of aesthetics or because we thought it was a nice game – why you should use the minimum of experimental data. The fact is, you remember, that we knew that Bragg and Kendrew and Perutz had been misled by the experimental data. And therefore every bit of experimental evidence we had got at one time we were prepared to throw away, because we said it may be misleading just the way that 5.1 reflection in alpha keratin was misleading. …They missed the alpha helix because of that reflection! You see. And the fact that they didn’t put the peptide bond in right. The point is that evidence can be unreliable, and therefore you should use as little of it as you can. And when we confront problems today, we’re in exactly the same situation.[15]
Crick is here emphasizing Plato’s point, I take it, that observation is not be entirely trusted. Observation alone is suspect.
The point was also made by Eddington: “It is also a good rule not to put too much confidence in the observational results that are put forward until they are confirmed by theory.”[16]
Another example would be the Law of Free Fall. If we were simply to drop things and measure their rates of fall exclusively by means of observation, as Galileo is supposed by some to have done, we would never derive the Law of Free Fall. The Law of Free Fall says freely falling bodies falling close to the earth all fall at the same rate of speed – disregarding friction and air resistance. But there always is friction and air resistance. Imagine, if you will, we are standing at the top of the Tower of Pisa. We drop, simultaneously, both a cannonball and a feather. These things will not drop at the same rate of speed. Observationally they fall at different rates of speed, and the observed event would be more in conformity with the opinion of Aristotle than that of Galileo. Galileo in particular was a scientist who had to argue very often and very forcefully against what would normally seem implied by the empirical, observational data.[17] As pointed out by Kuhn, Eddington, and Crick, observed “facts” are never incontrovertible. The association between theory and observation is a very fine tension in which either may be changed by the other, but that doesn’t keep science from still being primarily theory-driven.
Thus, we may have no true knowledge with observation alone. As pointed out by Plato, observation may produce opinion only, not knowledge, and is more obscure and less clear than knowledge. We generalize and believe as a matter of faith, opinion, or dogma, that all swans are white – until we observe black swans. Induction follows only as a matter of probability, not necessity, and generalizations drawn from observation may only result in induction. The instrument of truth is the theory, not the observation.
The Metaphysical and the Epistemological Dimensions of the Divided Line
Your interpretation is quite sufficient, I said. And now, answering to these four sections, assume these four affections occurring in the soul – intellection or reason (noesis) for the highest, understanding (dianoia) for the second, belief (pistis) for the third, and for the last, picture thinking or conjecture(eikasia) – and arrange them in a proportion, considering that they participate in clearness and precision in the same degree as their objects partake of truth and reality.
What Plato is saying here is that there is a direct, proportional ratio between the epistemological and the metaphysical dimensions of the world. In other words, the more reality or truth a thing has, the greater its potential to be understood by the mind; the less reality a thing has, the less it is capable of being understood. Sensory images, optical illusions, and generalizations drawn from these tell us very little. Understanding of essential structure is true knowledge, says Plato.
I shall bring to your attention that anyone who believes in abstract structural processes of a scientific, mathematical, economic or historical character would seem to be a metaphysical Platonist in some sense. It is knowledge of structure that renders function comprehensible. Anyone, however, who pursues analytic thinking and is willing to jettison that person’s unsubstantiated dogmas is an epistemological Platonist.
Conclusion
Again, I want to thank you for asking me to give this paper. The return to scholarship has been enjoyable and invigorating. Wisdom, I believe, is the ability to differentiate our happiness from our pleasure – and chose the former. We cannot do this without recognizing something not readily apparent: our actual happiness. According to Plato, the two roads to Larissa are that we may recognize our happiness by either blundering into it haphazardly or by using the method – not of induction or deduction – but of abduction or dialectic. This, in fact, is way philosophy and any other rational discipline has been pursued in the past: hypothesis, testing for contradiction, throwing out the hypothesis in favor of another suggestion that is an improvement over the first, deducing consequences, and continually testing the ultimate result. This is the method that results in true knowledge.
What I have done here is this. I have identified Plato’s dialectic as an account of a process that is also followed in modern day science – though I do not know why this should not have been pointed out before. I have rehearsed and accentuated Cornford’s explanation of why mathematics is included in the lower rather than the higher Forms – though I do not understand why this should have not been accepted or generally appreciated before now.
I therefore apologize for having occupied your time with the obvious.