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THEORY of FORMS: FURTHER DISCUSSION
Last week in class we attempted a discussion of the theory of Platonic forms. This is a brief summary of the ideas and points that were made in the discussion in class. The basic question is formulated in (A) below, and the content of our discussion in section (B) below. I supplement this with remarks of my own (section (C) below).
(A) THE QUESTION AT ISSUE
Very briefly, the argument in question is Plato’s argument for the existence of Forms, given in allegorical form in the discussion of ”the Cave” (see the ”SUPPLEMENTARY NOTES” on the Course Web Page). This argument typically starts from the observation that instances of justice, or of a ”circle” or ”square”, can be found all over the place. However, if we look at a square object it is not a defining instance of a square, but rather it possesses in approximate degree the quality of ”squareness”. Now Plato, mindful of what was already known in geometry and mathematics, argues that a perfect square exists nowhere in the world of appearances, but can nevertheless be defined mathematically. Not only can a perfect square never be found in Nature, but also any square we do find is only one of many imperfect examples, which hardly define for us a ”real” square. Such a definition may not need any elements of the real world for its definition, but nevertheless a square CAN be defined, and in this case Platoe argues it must exist. If so, there must be a ”higher realm” of ”forms” or ”Ideas” in which many such forms exist. In fact there will be a hierarchy of such forms, accessible only to reason rather than to the senses. Although this is not immediately relevant to the question at issue, we note that according to Plato, the highest (ie., most ”primitive” or ”fundamental”) of these was the form called ”the Good”, a kind of perfection to which all other ideas were subsidiary. Note that ideas or Forms, by their very nature, do not change (unlike the world of appearance); and anything that does change is thus not an Idea or Form. True knowledge can only be of Forms. This argument (sometimes called the ”Many to One” argument) is put in this way to emphasize the importance attached by Plato to mathematics- undoubtedly he was strongly influenced by the mystical ideas of Pythagoras as well as his concrete work in coming to this formulation. One can also view the argument as an attempt to start from language, in which names denote different specific objects, called ’particulars’, and then extend these features of language to argue for the existence of what are sometimes called ”universals”- a concept due to Aristotle. The idea of Plato is that if we are given a whole bunch of particular ’squares’, all of which are actually different but all of which have something in common, then this thing they have in common (what one might call ’squareness’), comes because they all to a greater or lesser degree resemble the perfect ’Square’, which exists in the world of Forms. Another way of looking at this is to say that if we have something like a particular square or a horse, then these are examples of squares and horses - but the only thing that makes them so is that they all have something in common. According to Plato, this thing that they have in common is that they resemble or in some way ’partake of’ the Forms ’Square’ and ’Horse’ respectively. Plato’s Theory of Forms is an example of a ”Theory” of the kind that many of the Greek philosophers were interested in giving - but it was a theory of very peculiar kind, and very different from most other theories put forward by the Greeks. It is useful to compare what Plato did with the work of the Atomists, and of Aristotle, to see the variety of theories that were proposed. In the same way it is interesting and useful to look at the astronomical work of the Greeks, which led to theories of cosmology - see in particular the work of Aristotle and the much later work of Ptolemy.
For more details go to the course notes. If you read more by Plato you will be able to explore his other arguments both for, and later on, against, the idea of ”Forms”. The key thing to understand here is the arguments, not the detailed history. Try constructing your own arguments - this is the first step to doing this kind of philosophy. As a useful exercise you can begin from the class discussion, summarized below.
(B) CLASS DIALOGUE
Here in (B) I summarize some of the issues and arguments that were brought up by various members of the class. In (B.1) I have tried to organize the arguments that were discussed, express them in a coherent way, and make a few remarks on them; in (B.2) I then remark on a number of points that were not raised but which are relevant to the discussion we had. In section (C) I go into more detail on some of the ideas noted in (B), and also a few related points, noting how
Plato might have responded to them, and pointing out a few other things you might like to think about.
B.1: POINTS RAISED in CLASS
There was a lot of uncertainty in class about how to take Plato’s argument - this is not surprising given that we live in very different times. Thus most of you did not have strong opinions on his theory. There were 3 main ideas that came out of the discussion, as follows:
(i) Different Kinds of Knowledge: Plato strongly emphasized that knowledge of reality meant knowledge of Forms, as opposed to knowledge of the world of appearances. This of course raised the question of different kinds of knowledge, of the degree of certainty one can have of different sorts of knowledge, and of the different ways that one acquires this knowledge. Plato himself believed that we came to knowledge of forms via a combination of rational thought and innate understanding. Some attention was given in class to the different kinds of knowledge that we possess. The different kinds of knowledge were divided into internal or ”introspective knowledge” of our own states of mind, visual and mental experiences, etc., and ”external knowledge” of the world around us. Finally, one can speak of knowledge of things that are not considered to be of the external or physical world at all - and this is in fact what Plato was particularly interested in. We can, for example, talk of ”unicorns”, generally agreed to be imaginary” as well as of abstract mathematical ideas like number. Plato viewed such knowledge as innate - not coming from experience - and included amongst this innate knowledge our facility for rational thought. Thus one can also introduce a distinction between innate and acquired knowledge. All of these distinctions are easy to criticize - but note that it is common for people to maintain that both introspective knowledge and ’innate’ knowledge are more reliable than knowledge gained from the external world. This opinion is of course also easy to criticize - we noted in class the existence of ”hallucinations”, and other mental experiences that were mistakenly supposed by those having them to refer to something real. It was also pointed out that often it is not so easy to decide whether we are referring to introspective knowledge or knowledge of the ”external world”. Thus, when we apparently experience - via our senses - events supposedly located outside us and independent of us, we nevertheless are apparently referring to experiences inside ourselves which lead us to infer the existence of an external world. However, one can ask in what way it is that experiences that are supposed to refer to things going on ”inside us” differ from experiences of an ”external world”. In what way are these different kinds of mental experience? A related point that arose in class was the degree of certainty that one can or does assign to different sorts of knowledge. What makes us decide that we are more sure of one kind of knowledge than another, or of one piece of information than another? One can approach this question by listing some set of different propositions, and then asking how much confidence one has in their truth; examples might be: (i) The sun will rise tomorrow (and old philosophical stock-in-trade) (ii) The sun will be shining next year (and we will see it shining if we go above the clouds) (iii) In the next football game (soccer in North America) to be played between Italy and New Zealand, Italy will win. (iv) In the next rugby game to be played between Italy and New Zealand, New Zealand will win. and one can add to this list - I will comment more on it below. In any case, the question arose in class - what sort of evidence did Plato have to justify his argument for the existence of forms? Indeed, what sort of evidence could one have, given that they do not exist in the world of appearances (what is often called the ’physical world’)? This raises important questions about what constitutes evidence for or against a given theory, and what makes one piece of evidence better than another. In this context it was remarked that our ideas on this have evolved enormously since Plato, and that we will return to this sort of question on numerous occasions, notably in the discussion of experimental evidence, of empirical philosophy, of Kant, and of 20th century philosophy of science. And most of all, in the discussion of modern physics
- ie., of quantum theory and General Relativity.
(ii) Are Forms just creations of Humans?: Another question that was raised concerned the specific limitations and special circumstances surrounding human knowledge. Clearly humans are not omniscient. There are clear limitations to our powers of perception, which although we are not directly aware of them, we know about by just by comparison with other animals. Although it is less clear to us, there are strong limitations to our mental capacities as well (this is less obvious to us because we are of course subject to them - we cannot think outside our own limitations). It is also clear that ideas or modes of thinking that are possible for one kind of mind may be impossible or ”unthinkable” for another. So given this, is there anything inevitable about any of the concepts or ways of thinking about the world that humans use?
forms, ad infinitum. Moreover one Form can be grouped into many different classes (eg., the number 2 is one of an infinite set of numbers, as well as being one of the even numbers, etc...). So how is one supposed to classify Forms?
(b) Appearance vs Reality - Experience and the ’Real World’: Clearly the most important point that Plato is trying to emphasize in his arguments is the distinction between the world we experience (the world of ’appearances’), and the ’underlying reality’, which we do not experience. However one can also ask: if we cannot experience this ’reality’, then how can we know anything about it (or even that it exists - see (i) above)?
(i) Rational argument as a source of Knowledge: Plato’s point of view is that we attain true knowledge by pure thought. This idea has played a huge role in subsequent thought. It acts as a philosophical underpinning for at least part of Christian dogma, and decisively influenced much of Western and Islamic philosophy. However the argument is seriously open to question. Certainly human history is littered with ideas, derived apparently by rational thought, which turn out to be pure fantasies, mere figments of our imagination. One of the two key features of the Renaissance, which led to a revolution against the Platonic way of thinking, was the rise of the ”experimental philosophy” (the other was the introduction of mathematics into physical theory). The experimental philosophy, as it was called by Bacon, Newton, and others, explicitly discussed the role of experiment in the discovery, testing and confirmation of new ideas about Nature - and it explicitly rejected the idea that facts about Nature could be demonstrated by pure thought.
(ii) Evidence for Existence of Forms: A related point to that made in (i) is this - if all our evidence for the existence of things comes from our experience, and Forms are not accessible to our experience, so that we cannot in principle have any evidence for them - then how can we know anything about them, or if they exist at all? Without concrete evidence, this argument says, we have no way of distinguishing fact from fantasy, as far as Forms are concerned. Note that by implication, this is true of any other mental object we think of which is not related to our experience. Those of you who were happy to dismiss Plato because he can’t have any evidence for his Forms may now feel a little uncomfortable. Which mental events are part of our experience, and which are not? Are all our mental events part of our experience? Or should we be making some sort of separation between ’objective’ experience, and ’subjective’ experience - and if so how? Try making a list of mental events or experiences, and decided which of these you are really sure of, and which not. To approach this point in another way, we can ask an apparently simple question: In what sense do Forms exist? Is it right to talk about them at all in the same way that we do when we talk of the existence of the physical world? Note that this is not an epistemological question about how we find out about Forms, or gain knowledge of them, but rather an ontological question about whether they exist at all - and if so, in what sense they exist. We can put it as follows, by simply asking: ”if they do exist, then where do they exist?”. In other words, ’where’ is this ”World of Forms”? Note that Plato himself was insufficiently clear on this - and as noted above, one can adopt the view that they may just be creations of the human mind.
(C) FURTHER REMARKS
In what follows I go over some of the points that are often raised in discussion of Plato’s Theory of Forms, and some of the answers given; and I make some remarks of my own. There is a clear overlap with some of the points discussed above, but I now extend this discussion to include other points of view.
(1) The Relationship between the World of Appearances and the World of Forms: This problem is often raised, even by those who think that Plato’s general ideas are more or less OK. Typical questions that arise are:
(i) We actually only know about individual ’particular’ phenomena that we observe; we have no direct access to the Forms, and in fact it seems highly unlikely, to most people, that the idea of Forms would ever occur without the particular kinds of experiences that we do have. From this point of view it seems that the Real World, from which the idea of Forms was extracted, is indeed composed of the objects that we experience. Thus, it seems unlikely that the idea of the form ’Circle’ would be conceivable without experience of approximate circles in the world of appearances. Some would go so far as to argue that if you can’t observe something in principle, then it doesn’t exist. So - how can we say that the world of Forms is more fundamental if this world is inconceivable (and maybe doesn’t even exist) except with reference to the world we experience?
(ii) Suppose it is true that the real world is not a world of Forms, and that there is some more fundamental world of Forms. Then in this case - what is the relation between the 2 worlds? Note also that a similar but more extreme problem arose with the discussion of Parmenides, who dismissed the real world of sense perceptions as quite illusory, and not corresponding in fact to anything ’real’ at all. Thus one can ask- in what way are Forms actually embodied in our world?
(iii) A final objection to Plato’s ideas on the relationship between the 2 worlds follows on from (ii) above. It arises again because Plato was not sufficiently specific about the relation between Forms in his inaccessible worlds, and the phenomena in our perceived world. It is then not only unclear how to understand the real world in terms of Forms- it is also unclear how we are to decide, by doing things in the world of perception, on the truth or otherwise of our ideas about the world of Forms. To put it in modern language- the theory of Forms is UNTESTABLE, because we can only test the truth or otherwise of a theory by operations performed in the real world. So how do we test ideas about the world of Forms?
Remarks: It is certainly true that Plato is not too clear on this general issue, because although he talks a lot about the world of Forms, he does not say so much about the real physical world! However, one suspects that Plato’s response to this would have started with his usual line of argument, viz., that whereas one cannot apprehend the Forms by sense perception, or by any inspection of the real world, they can be examined and understood- at least to some degree- by imperfect mortals, using the intellect, ie., using rational thought. Here of course his 2 guiding lights were (i) the ideas of mathematics, and (ii) the Socratic method of exploring and refining concepts, to isolate and extract their essential meaning. In fact Plato did try occasionally to talk about the real world of sense perception, and was rather interested in a number of different features of this world- notably in astronomical phenomena, and in the mechanisms of sense perception itself (for which he had an elaborate theory); see in particular the Timaeus. He also had a kind of theory of physics, which was based essentially on Pythagorean forms and the Elements of Empedocles- although Plato was very firm that the universe was in some sense alive, and that it had been molded by a ’Demiurge’. Concerning the objection that we cannot possibly know about forms in the absence of experience of the world of appearances, Plato argued that we do have innate knowledge of Forms - indeed, it would have been hard for him not to, in the face of this objection. In a well-known passage in the Meno Socrates coaxes the solution of a geometric problem out of a boy with no previous knowledge of the subject. Plato’s argument here was rather peculiar, and the details are not so relevant here (he argued that the boy’s innate understanding of geometry came because he had am immortal soul, and that it was the soul that actually had memory of geometrical ideas; but that true understanding of these could only come through ratiocination). The point is that it is hard to avoid the idea that some sort of innate ”geometrical understanding” on our part is involved in the discussion of geometrical forms. It has to be admitted that none of this makes it very clear in what way objects that we observe in the world of appearances are related to Forms, or how they ’partake’ of them. This point is made more acute when we ask how one might test Plato’s idea (which means of course a test in our world of appearances). By putting this objection in modern terms we make it harder to understand how Plato might have responded. He certainly would have objected that one can test the propositions of mathematics by purely rational means- by ’tests’ performed within mathematics itself, by purely logical manoeuvres. But this does not really answer the argument. Actually Plato did have a theory of the universe, and of how it is constructed (although it is important to note that, in the same way as Parmenides and Democritus, he did not pretend that he was sure it was correct). Given that, for example, his theory of how matter was constructed ultimately reduced to a consideration of geometric forms (the 5 regular or ’Platonic’ solids), one can ask what method could, in his theory, give us certain knowledge of its truth (or otherwise)? To many people in the 21st century, it seems clear that one cannot have certain knowledge of the phenomena of the perceived world, and that it is therefore not clear how it is possible to have any sure knowledge of the world of Forms, or indeed of anything at all. However many mathematicians have argued that at least in logic and mathematics this is not the case - that one can indeed establish mathematical or logical truths, at the severe price that one is dealing only with formally defined objects. The objects of mathematics, in this view, are not so different from the Forms of Plato. Thus his point of view has had a huge and enduring influence. However this still leaves us with the problem of making the link between formal mathematical objects and objects in the real world. We shall see later on that this problem is alive and particularly acute at the present time, because of quantum mechanics (discovered in 1925). We will also see that the whole question of what constitutes mathematical truth was turned on its head in 1931. Finally, one can ask to what extent one can even unambiguously relate objects in the real world to Forms at all. Thus, eg., the idea of all horses sharing some property or properties is all very well. But in some cases we can’t just define something by listing all its defining qualities or properties (as is done in a dictionary). This may be insufficient,
sciences). In particular it led them to confusion over the relationship between the objects of perception and the objects of mathematics. It is hard to guess how Plato would have reacted to questions about how the limitations of language might have constrained our understanding. This is because the study of language and logic was hardly begun at this time. We will come back to this question later, particularly in the discussion of modern quantum physics.
(3) Pre-conditions of Rational Thinking: A quite different set of points has much more to do with the purely logical features of Plato’s argument, and the characterization of knowledge of the ’Forms’ as entirely based on ratiocination. In particular:
(i) It is clear that exactly the way we categorize our experiences will depend on the structure of our perceptual and thinking systems, both of which are very limited. But surely some kind of categorization is necessary - and it is certainly quite hard to see how we can avoid very elementary distinctions like that between particulars and universals (cf. Aristotle), between subject and object, or between appearance and reality. So this leads to the question: How can we understand anything if we don’t have a theory of Forms (or some other metaphysical theory which allows us to understand our experiences)? (ii) Plato attempts to present the case for the theory of Forms in a way which is supposed to convince us that it is logically inevitable. However his ’derivation’ of the theory is presented in the form of a conversation, and one can see all sorts of points on the argument that can be queried. Plato (and many generations subsequent to him) were very impressed by mathematical derivations; and Aristotle set the whole idea of formal deductive arguments in motion by inventing the notion of a formal approach to logic. So, with the benefit of our modern understanding of logic: Can we rigourously ”derive” the Theory of Forms?
Remarks: The first question is actually of some importance for the philosophy of science (and in fact for philosophy in general). Let’s just focus on science, and to see why it is relevant to modern science, let’s rephrase it in the following way. It is clear that what we currently know of the world around us is based on a careful combination of experiment and theory. In the modern view, experiments are done which test theoretical arguments, and if the theory doesn’t work, then we need another theory. It is the experiments that are the ’bedrock’, giving us access to the truth, not the theory. But this view (and I have deliberately caricatured it, in a way which, unfortunately, many scientists would like it said) is really so oversimplified that it is very misleading. It is quite clear that any description of an experiment in physics or biology is quite meaningless unless one understands and accepts a lot of preliminary theory already - and this theory presupposes already a particular point of view on what is physically real. We will see this in much more detail as we go along in this course. Thus the ’categorization of the world’ mentioned above, and a certain theoretical framework built on top of it, is already there before one can even talk meaningfully about an experiment. Experiments are simply operations in the real world as well, and therefore, in our thinking at least, they are necessarily a subsidiary part of this framework. That this must be the case was apparently clear to Democritus but perhaps less so to Plato. Democritus reckoned our perceptual and mental apparatus to be connected, and to be simply another complex construction form the atoms. From this he drew the conclusion that our thinking and perception were necessarily limited and imperfect. This conclusion he would presumably have applied to our rational thinking as well - and so would have been forced to the conclusion that our ideas about ”Forms” were just as limited as our perceptions. If we accept this point of view, then it seems fairly clear that we are driven to the conclusion that ideas like ”Circle”, and the very idea that circles exist more or less approximately in Nature, are logically contingent on some accidental limiting characteristics of our perceptual/mental apparatus. There is then no reason to suppose that such entities exist - it is merely a way humans have invented to interpret the world. This argument suggests that all of mathematics and logic is nothing but some human invention. And yet many mathematicians would disagree - their point if view is more Platonic, and they feel that they really are discovering things when doing mathematics, not inventing them. This argument is complex. Note that even if we do accept the argument above, we are still assuming that there is some ”Reality” there, that we are part of, and that we are imperfectly aware of, and trying to discover. We shall be returning again to all of this.
Consider now the second question - whether one can derive or otherwise verify the Theory of Forms. Again, a modern discussion of this would lean heavily on our more sophisticated understanding of language and logic. Certainly we would still make the distinction between particular instances of, eg., a horse, and the general category of horses - but
the way this is done now is rather different. In modern formal logic one would simply start by defining the set of all horses, and discuss this set as a formal object in itself. Then a ’derivation’ of the theory of forms would take us into the discussion of how formal set theory is constructed, and how theorems in set theory are derived. Without going into this at all, I simply remark here that while these discussions may not seem to have any relevance to physics at all, yet again we will see that in the context of modern physics (particulary quantum physics) we will have to think about them again. Curiously - and this would no doubt have upset Plato - modern physics, which is very heavily tied up with mathematics, has brought the idea of a derivation of ideas in mathematics somewhat closer to physics. This is because nowadays, our description of the world is very complex mathematically - and the most important entities in modern physics, like fields, or the spacetime metric, or the quantum probability amplitude, or not in any way observable directly, even in principle. They are simply abstract entities like Plato’s Forms, simply because we can find out about them, by using their behaviour (characterized mathematically) to make predictions about the real world of appearances, and then compare these to experiment. The connection between these ”forms’ and the real world of appearances is also embodied in the theory - in fact the real world, including ourselves, is built out of these entities. This is fundamentally different from Plato - who, as we already discussed in (1) and (2) above, was not clear on how Forms related to the world of appearances.
(4) The Ontological Status of Forms: A serious objection that can be raised against Plato is that he is making a rather important assumption in his theory, which is that if we can talk about a general property like ’squareness’, there must always be an object corresponding to this property, to which the general concept of squareness must be referred. This assumption is crucial to the theory of Forms: he assumes that there must be a ”Square” Form, in the world of Forms, to which particular approximate squares in our world are related. There are 2 obvious objections to this assumption, viz., (i) Just because we have some quality, does not mean that there must be an object to which we must refer in defining it. Thus if we say that certain things are ’heavy’, it is in no way clear that there must be some ”Heavy’, or ’heaviness’, or even ’Weight’, from which the heaviness or weight of given objects is defined. To summarize- there is no obvious reason why qualities have to be converted to special ’objects’ called Forms. One might say that Plato is making an elementary confusion between 2 categories, of objects and qualities (ie., he is assuming that all adjectives must have a corresponding noun). (ii) Even if one is prepared to accept that to all qualities and other sorts of abstract concept there must be a corresponding object, or ’Form’, there is another hidden assumption, viz., that just because we can conceive of, or discuss something, it must thereby exist. This is certainly not obvious. For example, do the objects in a hallucinatory experience exist? To recall the example of Bertrand Russell, does the ’King of France’ exist? One can clearly conceive of many things (unicorns, etc.) that nowhere exist in the universe. Now if we accept that the existence of an ’idea’ in someone’s head does not necessarily imply that there is anything existing anywhere, in any realm, to which this corresponds, then one can surely argue that Plato is wrong to assert that the existence of ’circles’ in the real world implies a ’Circle’ in a higher realm of Forms.
Remarks: It is not obvious how Plato would have treated the first objection here- again, it turns to a great extent on the grammatical structure of language and on our notion of logic. One suspects that Plato would have fairly quickly dealt with the second objection, as follows. Again, the fact that, eg., no real unicorn exists is irrelevant to his argument- for Plato, one can easily have a Form with no exemplars in the ’real’ world. Not only is no contradiction involved, this is actually a fairly natural consequence of his idea that the real world is a highly imperfect, ’dumbed down’ correspondent to the world of Forms. However, he would argue, if we can conceive of or imagine something, then the only possible way that this can be is if the properties or qualities of the imagined object refer to some ideal qualities in the world of Forms.
(5) Limiting Processes and the Theory of Forms: A we have seen, one view of the ontological status of Forms is that they might well just be creations of the human mind - that there was no reason to assert that they really existed ”out there” somewhere. Thus, if one could view the different approximately circular objects in the world as imperfect representations of a ”Circle”, then one is not forced to argue for the existence of a real Circle - instead one could just think of it as an idealized limit, in the sense that objects in the real world can be viewed as exemplars of a set of objects which ’tend towards’ the limiting case of a perfect circle. This argument is similar to the idea of, eg., a
