Joe MacMaster, St. Francis Xavier University

4D Forms as a Solution to the Problem of Forms and Particulars Joe MacMaster, St. Francis Xavier University

Arguably Plato’s most significant contribution to the world of philosophy is his theory of the forms. These forms are mysterious things which can only be grasped by the intellect as opposed to any sense perception (Timaeus, 28a). Though mysterious, and the basis of much questioning and philosophical discussion, the forms pervade every aspect of our existence. In Phaedo Socrates teaches that after death the philosopher’s soul goes on to the world of the forms (63e-64a). In the Republic Plato uses the Divided Line analogy to explain how the forms are both beyond, and more real than this world (509d-511c). In Timaeus we are introduced to the Receptacle of all Becoming (48e - 49a). In this paper I will argue that for the theory of forms to work, the forms themselves must be four-dimensional objects that intersect our three-dimensional world.

My initial reasoning for why this must be the case was to solve the problem presented in Parmenides that if forms can be expressed in different unconnected locations at once, then a form is separate from itself, or that a form is not One. Forms being either separate from themselves, or not One, is contrary to the theory of the forms that Plato seems to posit throughout his works. The 4D form model allows for a form to exist independently, undivided from itself, while also being able to intersect all points of the 3D world without compromising its nature. This theory faces several challenges. A) It is unclear whether this argument works just as an analogy or if it actually how the forms relate to particulars. If it is just a helpful analogy, where exactly does it break down, and what exactly does it explain? B) If the forms actually are 4D objects, what’s to say that two different 4D objects couldn’t appear exactly the same to us as they intersected the 3D world. It seems like the forms must be extremely different from the things we see in the material world for this to be the case either by analogy or literally.

In this paper I will argue that the 4D form theory in not just an analogy, but that the forms are in fact 4D objects. I will break this argument into four parts: (1) The necessity of forms as having more than three dimensions. (2) Why this theory must work literally and not just as an analogy. (3) How the forms retain their essence while intersecting this world. (4) Our understanding of the forms is limited

to our understanding of their effects, and so they may be radically different, higher order actualizations of what we see them as. This view does not arise explicitly from Plato’s beliefs, but based on that nature of the forms that we get from Plato I believe this is the most adequate explanation to solve the problem of forms and particulars. After making these arguments I will address several other problems and considerations that may arise throughout the argument.

1. Why the forms cannot be limited to the third dimension

In Parmenides, Socrates is asked how the forms relate to particulars. Parmenides himself raises the problem that if an object possesses a certain form, it needs to be clarified whether or not it possesses only a part of the form, or the whole form (131a). If it possesses the whole form, then it seems that the forms are not one, because a certain form can be found in many different places. If the full form of X was found in many different places, then there must not be just one of X. Plato’s works in general lean more towards the idea that particulars partake of only a part of a form, and not that each particular possesses an entire form or many entire forms, which defeats the idea that they are one-over-many. The problem of how a form can be found throughout many separate places in space at one time and yet still remain one and be undivided from itself still remains. This is where I believe the 4D form theory becomes necessary.

If a form exists entirely within the third dimension, and is bound by 3D laws, then seeing a form appear in multiple places at once would mean it is separate from itself. A form cannot be separate from itself, since oneness and indivisibility are properties that each from has (Parmenides, 129b, 131a-b). Therefore, the forms must not be bound to the laws of 3D objects, and do not only exist in this dimension. The forms cannot be of a lower dimension, since that would make their expression in 3D objects impossible, so they must exist in a higher dimension. The problem that exists with discussing objects of a higher dimension in space is that they are physically impossible for us to visualise or grasp. We can only understand how objects of a higher spatial dimension relate to objects in our own dimension. This is done by observing how objects within our own 3D world would interact with objects in a 2D or 1D world.

2. 4D forms are literal, not analogical

Something that must be clarified with this idea is whether or not it is to be interpreted literally or as an analogy. I think it must be taken literally, since if it is just an analogy, the forms must only exist within our dimension, or within conceptual space. I do not think conceptual space serves as a complete model here, since the existence of objects in conceptual space depends on human thought.

Since the forms need to have existed prior to, and independent of human thought, I believe they must have existed outside of conceptual space. We are left with no choice but to accept that the forms reside in a higher spatial dimension. A problem here is that we cannot mentally grasp or visualize a 4D object. What we know about these objects and how they relate to our 3D world is based entirely on how 3D objects would relate to a 2D world, and how 2D objects relate to a 1D world. It is worth noting that this 4th dimension is not referring to time, but another spatial dimension that is completely invisible to us. We are used to using an x, y, and z axis to describe 3D space. A 4D space would have a w, x, y, and z axis, as well as time. Just as the z axis would be invisible to any sentient being in a 2D world, the w axis is completely invisible to us, but it would be the axis that 4D objects would move along.

In a previous paper I used the analogy of a tuning fork intersecting a 2D world to illustrate how 4D forms would intersect our world. To recap, if we came upon a 2D world with sentient beings in it, and inserted a tuning fork into that world, the beings there would see two separate circles appearing out of nowhere. Since they cannot detect what is outside of their 2D world, they would be unable to see the rest of the tuning fork, and so the two prongs of the tuning fork would appear as separate, but virtually identical objects. This analogy seems to illustrate how the forms literally relate to our world. Take the idea of an empty 2D world and assume that there is some creator who has to make a world in this empty plane of existence. This creator is a 3D being, and so the materials available to them are also 3D. These materials can intersect the 2D world, and so the creator builds a world from these 3D objects along the 2D plane. Even though what gives existence to everything within that world now is truly a 3D object, whatever exists and has sentience within that 2D world is only seeing a “slice” of those 3D objects, completely void of depth. The 3D objects are analogous to the forms, and those sentient beings are analogous to us. This is a highly simplified analogy to how a 4D form network would weave the reality of what we see in our world. But just as Plato believes that it is the forms that are more real than us, those 3D objects are more real than what they are perceived as by the 2D beings.

The next problem that arises from this is that if forms are intersecting our world and we are perceiving them, how can we tell them apart? Going back to the 2D world analogy, I could insert a tuning fork, a radio antenna, and a stainless-steel knitting needle (all of equal diameter and external appearance) into that world. Each of them would appear as identical objects to whatever beings were looking at them, despite the fact that they are all entirely different objects with different functions.

How do we know that the same does not happen with the forms within our world? To answer that we must examine the nature and essence of the forms.

3. Forms must always retain their essence, which is pure.

To remove the essence of an object is to remove the object entirely. In the scope of the forms, a physical object may not be limited to just one essence, if it is the forms that give a physical object its essence. The issue of purity gets complicated in the discussion of how forms partake of other forms. Despite that, I believe it is safe to say that the forms as objects are of radically different substance and character that mere 3D physical objects. Plato concedes that the things in the physical world do not necessarily have one forms associated with them. In Parmenides we see Socrates questioned as to whether there is a form of hair, mud, or dirt (130c). If physical objects get their nature from one form, then there must be a form for every physical object. But in that passage, we see that not every physical object has its own form, and so we know that an object may partake of several different forms to become what it is. This composite action of the forms seen within the particular realities of this world brings me back to the previous problem mentioned. How can we be sure that two distinct forms would not appear as the same thing when intersecting our world? To answer that I will bring to light the essence of the forms. If forms are not changed essentially when objects partake of them, then they retain their unique essence within this world. A form’s essence is not changed into something different when an earthly object partakes of it (excluding Cambridge Changes, which do not alter the nature of an object). By this I mean that a from does not become a different form because something partakes of it. Just as a tuning fork that is intersecting a 2D world does not become not-a-tuning-fork, form X does not become ¬X when it intersects this world. Therefore, a form retains its essence in this (the 3D) world.

The key difference between a form and a tuning fork is that forms don’t have parts in the same way a tuning fork (or any physical object) does. I can’t speak about the handle, or the left and right prongs of a form. Now it may be the case that the 4D form has some sort of 4D “net” shape, which would allow it to intersect our universe all over the place. But the form itself is of identical substance and essence at one end of the earth where it intersects as it is at the other end of the earth and every other point on its body. Because of this pure unchanging essence throughout the form, and the self-perpetuation of forms, each form would be unique in the way it intersects our world. One thing that the aforementioned tuning fork, radio antenna, and knitting needle share is a material. They’re all made of shiny stainless steel, and in reality, are themselves composites of different parts and materials with

different functions. If each form is something unique, without duplicates, then forms will not have identical extrinsic appearances, and thus we will not find two identical things coming from different forms.

The nature of the forms must be such that at any point that they intersect our world they give off the same appearance to us. For example, wherever the form of the Circle intersects, or whenever an object partakes of the Circle, we see that object as circular. Likewise, the form of Two or Three will always yield something we perceive as two or three. But as we know, what we see in the physical world is always an imperfect version of the higher form associated with it (Republic, 509d-510a). This leaves us to resolve how something perfect can come across as imperfect, but also how something so grand, complex, and able to intersect our world at every given point can convey something as simple as a square. I think the resolution lies in recognizing that there is a degree or equivocation between a property x and the form X. We know that the form gives rise to the property, but just how similar is the form to the particular?

4. The nature of the forms may be radically different than our perceptions.

Since we can only grasp the effects of 4D objects and not their essence, we can never understand the pure essence of these forms. They are a mystery to us so long as we are bound by human flesh in this world. We can only hypothesize as to what the forms in the higher realm look like. This does not necessarily mean we can know nothing about the nature and behavior of the forms. Going back to the tuning fork example, what would be seen in that 2D reality when the tuning fork was inserted is radically different than the nature of the entire object. There is the fact that matter as we know it is fundamentally three dimensional. If something 3D is inserted into a 2D world, only two dimensions are perceptible. There is height and width, but absolutely zero depth. While the essence of the entire object is not altered, what is being perceived in that 2D world is something fundamentally different than what the object really is.

Recall the Divided Line, and the Allegory of the Cave, where Plato implies that intelligible things are imperfect copies of something higher. I think that makes perfect sense in light of this theory. Again, the difference between our physical objects and those in the world of the forms is that the forms are of a pure essence and are self-perpetuating. It is probably the case that if we could mentally grasp the form of X to its full extent, it would be something radically different than the x that we are used to. The four-dimensional element of the forms is not analogical, however our interpretations of them probably are. What we call “good” may be an analogy to what the form the Good is. In such a way that of we could comprehend the form of the Good fully we would be comprehending something we never

understood existed. If we could experience the form of the Good with all aspects of our being, we may realize that what we call “good” barely compares to how good the Good is.

I frequently describe the forms in the four-dimension using three dimensional terms. The use of these terms like describing a “form net” arise due to a lack of appropriate language. As beings limited to three dimensions, our way of describing things does not extend beyond three dimensions. The best way to describe the shape of the forms may be a sort of “hyper-net” in which all of the forms can reach all parts of our universe. V. Harte’s paper on Plato’s Metaphysics cautions us to avoid viewing Plato as preaching two separate realities (Harte, 14). In this model of the forms, a single reality is still retained. It is not as though the fourth dimension where the forms reside is some alternate reality, but rather an extension of our universe inaccessible to us in this life. We are still left with several problems and considerations, which are discussed in the next section.

Additional problems and considerations

After establishing my thoughts on the limitations of 3D forms, the literal aspect of this, the retention of essence in this dimension, and the likelihood of forms appearing as radically different to us in contrast to their 4D nature, I see several problems still unanswered. (I) Until now the examples of the forms given, and the general forms in question have been physical, abstract things. Forms like those of certain shapes, numbers, and material properties. These types of forms make sense using a physical, or “hyper-physical” model of forms to explain, but the model may seem to omit forms that have no physical or abstract properties. The forms of Justice, Piety, The Good, The Beautiful, Being, and all others like this seem difficult to reconcile with a model that uses 3D physical objects as an analogy. (II) Are the forms limited to only the fourth dimension, could there perhaps be forms that possess five or more dimensions? (III) Why do the forms intersect the world where they do? Platonic writing often speaks of things partaking of the forms, but if you remove all of the forms that the object partakes of, what are you left with? (IV) If physical objects are a series of form copies, then they do not actually have the form, but are just modeled after the form. This seems contradictory to the previous partaking notion we see in Plato’s earlier works.

The model for the forms that I have proposed suggests that the forms are 4D objects that penetrate our 3D world weaving the intelligible aspects of reality. This works well for the forms that give rise to physical properties, but not all forms do. In Euthyphro the dialogue works around determining the true meaning of Piety. In

Republic the dialogue opens up with a discussion on the true meaning of Justice. If the discussions were on “what is a Triangle”, or “what are Odd and Even” the discussion may have been shorter. I believe that is quite difficult to fully describe the majority of forms though. Even something as simple as Two seems impossible to explain without the use of other numbers, which themselves are forms. Despite this, we still have a functional understanding of Two, and other simple positive integers. The same is the case with forms like Justice, Beauty, and Piety. We can’t simply define their full and true essence, but without a functional understanding and ability to recognize these things, we would not even be trying to understand them. If we didn’t understand a single about the forms in question, we would not be able to discuss them, or recognize them. But we discuss these forms because they are patterns or relations that we recognize and understand, albeit an imperfect understanding perhaps.

The forms like Justice, Piety, Beauty, and even Tallness, have no intrinsic physical properties themselves. What they are seems to be relations. Justice deals with the interactions and how people behave towards one another. Piety is a relation has with those of higher honour, such as a father or a god. Beauty seems even harder to explain, but it is one of those things that we can immediately recognize when confronted with it. Tallness is the most obvious as being a relation, what is tall in relation to one thing is short in relation to another (with two exceptions, presumably the single tallest, and single shortest thing in the world). If these forms are forms of relations, then the forms themselves must be that of pure relation. Not just Relation itself, but of the particular relation that the form in question is. The relations that Justice and Piety are must be present and expressed to the fullest extent at one extreme of their form as they are in another extreme. Where we see Justice or Piety expressed, the form of either must be present, having intersected our world in the locations of their expression in some way.

So far, I have not addressed the extent of how high the extent of the forms may go in terms of dimensions. If the forms exist in four dimensions, could they also exist in five dimensions? Due to our limitation to only knowing three dimensions, it seems difficult to know for certain, but I do have a thought on this. There are some forms which transcend the others. The form of the Good, Being, One, and any other form that seems to be the cause of other forms, must permeate throughout the world of the forms in such a way that those forms can partake of them. It would make sense that forms like this may occupy five dimensions and interact with the 4D forms in a similar way to how the 4D forms interact with us. If there is any form from which all other forms proceed, this form would occupy the

highest number of dimensions. As to how high this dimension is, I see no reason for it to be any higher than necessary. We occupy three dimensions, we are not limited to two, and do not have four. This does not seem to be out of any obvious necessity and is a simple mode of existence considering the number of dimensions we occupy could be far greater and complex. Because of this, based on purely inductive reasoning, I suspect the forms must also occupy the simplest realm that they can.

For any given object, an array of forms can be identified in it, or it could be said that it partakes in an array of forms. Even the four platonic elements, fire, air, earth, and water have several forms attributed to them in their pure state. As I write this, I am employing the use of a table. My table, and any given table must partake in a series of forms in order to give it it’s properties. Maybe there is a form of table itself, but without other more fundamental forms a table could not come to be. A table requires rigidity or hardness, and a degree of tallness in order to raise it off the ground. It will have some sort of colour, and will employ certain shapes (in my case, a square). If this table were to cease partaking in certain forms, what would become of it? If it stopped partaking in the form of the Solid it would collapse, if it stopped partaking in the form of the Square what shape would we be left with? If these forms and their properties were entirely removed, would we be left with anything at all? Additionally, it seems that it was human craftsmen that instilled these forms into the table to some degree. If a carpenter carves a square shape into some amorphous piece of rough wood, does the carpenter then have some bearing on the form of the Square? It appears as though we can move the forms in some way, such that they manifest themselves to a degree within our world, by our doing.

In the Timaeus we are introduced to re Receptacle of Becoming (48e-49a). From this, we see Plato’s idea that the elements are qualities, not substances. When we speak of fire, we ought unchanging stability, whereas “such” implies a quality temporally (49d-50a). Perhaps what we are left with after we strip away all of the properties or forms from an object is the Receptacle. The Receptacle must be what allows physical objects to partake of one form, and then another. If you mold a clay brick into a sphere, you are changing the form that the clay partakes of, and it is the Receptacle that allows this change, and for the objects to partake of any forms in the first place.

In the Timaeus we see the idea that the Divine Craftsman creates a perfect image or model of things in order to base physical creation off of (30c). If physical objects are just copies of the forms, and the forms just a model to base everything else off of, then how can particulars be said to partake of the forms? If I sketch an image of an apple, my sketch could be called a copy of the apple, which is the model. Even though my sketch is an imperfect representation of what the apple is, I can say that it was modelled after the apple. Despite that, I would not say that it partakes of the apple, since its existence is independent of the apple. Though the forms cannot be destroyed or changed as the apple could, the copies in both cases do not actually have any physical share in the model themselves. This is where I believe the 4D theory can preserve the idea of particulars actual partaking, and not just being modelled after the form as an image. If everything, including those things like justice and piety are just counterfeits of the true version, then the forms aren’t so much the cause of our being so much as the Divine Craftsman is, though he just uses the forms as the model to base his creation off of.

In review, I have discussed why the forms cannot be limited to three dimensions, why the 4D form model must be literal and not analogical, how forms remain unchanged objects when objects partake of them, and how our understanding of forms is derived from understanding their effects. In conclusion, I believe the best explanation for how the forms relate to particulars is through the 4D model. Though nowhere in Plato’s works did he suggest this explicitly, the way they are described in much of his work, in combination with the problems raised for the forms in the Parmenides, leads me to believe that he may have been open to the idea had it been presented to him.

Plato characterized the forms as being intelligible, but not perceptible. That is, we can come to know them only through philosophical understanding, but not through any sort of sense perception. We cannot gaze upon a form, or hear, or smell, or touch a form in its pure state (Republic 78c10–79a5). I think this coincides with the idea of a 4D form. By their nature, they are simply impossible for us to sense. Our sight is limited to 3D light, as well as the objects we touch, and the sound waves we hear. Our senses are at a disability to perceiving this type of object, but the mind is not as limited. Through sense perception we come to learn about things, and through philosophy we can develop our understanding beyond what we physically perceive. Such is the case with this model, which may serve as a solution to a very old problem.

Cooper, J.M., ed. (1997). Plato: Complete Works, (Indianapolis: Hackett Publishing Company) .

Harte, Verity (2008) Plato’s Metaphysics, The Oxford Handbook of Plato, 1 edition.