# My professor claimed that "F = ma" is a Platonic form. I disagreed, but is it?

My professor claimed in a lecture that equations such as "Force = mass x acceleration" are Platonic forms. I disagree for a few reasons:

A form is non-spatiotemporal. Force is a vector, which has magnitude and direction. In mathematics, vectors describe space.

Force = dp/dt. That is, it is the change in momentum per change in time. It is dependent on time.

Furthermore, a force is independent of human reality. Newton's equation is dependent on the use of the Euclidean metric. But on another planet, that society could use a completely different metric. For instance, the discrete metric or the taxicab metric, which would make their equation for force completely different.

He's also said some other irksome things, such as the Pythagorean Theorem (a^2 + b^2 = c^2) is a Platonic form. I also disagree with this one.

The Pythagorean Theorem depends on the acceptance of Euclid's Fifth postulate - the parallel postulate. But there is not universal agreement that our universe is best described by Euclidean geometry.

There are competing geometries such as Lobachevskian geometry (geometry on a sphere) or Riemannian geometry. These are gained by taking the negation of Euclid's postulate.

So I don't understand how equations such as these can be Platonic forms - that is, non-spatiotemporal ideals independent of human realities.

You used an example like Newton's law.

It may be viewed as a formula expressing a causal connection between "natural" facts : force (acting on bodies) and acceleration (change of velocity, that is - according to Newton - change of "status").

But, with the development of physical theory, it can be viewed as a theorem proved from "axioms" or more general principles of physical theory.

In the first case, it is difficul to relate it to Plato's conceptions. It can be viewed (as Hume do) as an "empirical generalization" that can be inductively supported but that - in principle - can be confuted with the progress of knowledge and with the aid of new experiments.

In the second case, it can be treated as a mathematical theorem : from arithmetics, like Pythagorean theorem, or one of the theorems of Euclidean geometry.

A new comment : arithmetics for Pythagora and ancient Greeks were very very different from ours. Numbers were only natural numbers, what we use for counting, and the discovery of irrationals (square root of 2 = lenght of the diagonal of the square with side 1) forced them to develop axiomatic geometry including a general theory of ratios of magnitudes, where magnitudes were not (for Greeks) numbers.

So, the issue is about the relation of a theorem of mathematics (in axiomatic form) to natural facts : e.g. the parallel axiom is "applicable" to our physical space ?

This is the point where Plato "fits" : the relation between "ideal" concepts and truths to the "natural" (empirical) facts and experiences.

The form of the circle is an ideal (but a real existing one) to which physical circles must be "compared" : they incarnate it, they are copy of it, they ...

The usual metaphor is with a chair : all physical chairs are different, but they share a common form (but which : four legs ? the function of being used to sit on them ? what is the abstract "chairness" they share ?); or with moral and esthetical concepts: the Good or the Beatiful.

Going back to geometry, the physical straight lines (a stick) are copies of the ideal platonic line: but a stick is not without thickness, it is not perfectly straight, ...

We can adopt a "moderate" platonic attitude, like the one you can find in Galileo: yes, the language of nature is mathematical, and if we find an iron ball that is not "perfectly" spherical, the problem is not that it "incarnates" the sphericity in an imperfect way: a sphere is a sphere. If it is not perfect, it is only due to the fact that its shape has a geometrical shape that is different: a more complex shape, that needs a more complex "equation" to be described.

A sphere in the natural world is as much spherical as a perfect one ... if there is one. Otherwise is not a sphere at all, but only a more complex geometrical shspe.

So we use approximations, and our calculations works well.

So, what is the conclusion ?

Plato is still worth to be read for a lot of reasons; Plato's conception of a realm of "pure", "abstract" existing thing (but immaterial) is still appealing in some field, like philosophy of mathematics, but to say that a physical law is a platonic form, without other clarifications, it semmes to me of little usage for a correct understanding of a deep and complex issue like the nature of scientific laws and the applicability of mathematics to the natural world.

As with everything, it depends on what we mean by "Platonic form" and F=ma. If we mean a form that Plato himself articulated, then clearly F=ma and a^2 + b^2 = c^2 are not forms. If we want to be more charitable in our interpretation, we can presume that we mean that they would qualify under the rules Plato has for forms.

Here, I think there's good reason to think your professor is right to call them forms. In your question, you explain what a force is, relating that forces only occur in time, and that forces are independent of human reality. I will address the last point first. For Plato, forms do not depend on human reality. They are not derived from our experience and exist regardless of our existence. So this is a point in favor of your professor's claim -- not against.

Moving to the first half, existing forces only occur within time as changes in momentum. But this actually doesn't prove that F=ma is not a form. F=ma is an equation that describes the relationship between mass and acceleration, but it is not itself F, m, or a. To illustrate,

it takes a force of 4 Newtons to accelerate 1 kg ball 4 m/s^2. So the force = 4 N. The mass = 1 kg. The acceleration is = 4 m / s^2. But where is the F=ma? It is the relation that holds between them. But if we think this is an unchanging law, then it's quite similar to a form. We generally understand that as empirically discovered, but for Plato reality is recalled per the Meno and Phaedo. And what is recalled and reimplemented is the forms. So if this is a fundamental law that is untainted by the world, I don't see why it can't be a form.

The evidence against your professor's claim seems better for the denial of the Pythagorean theorem. Namely, if we can prove something isn't true or is merely contingently true, then its not clear how it can be a form. The hard part will be figuring out whether things like mathematical relations are so.

Let's explore your teacher's irksome claim.

1) According to Plato, which is what your teacher is claiming, there are Forms of beauty, mountains, colors, courage, love, justice, goodness, etc.

2) Therefore, according to Plato, there are Forms of objects (like mountain) and attributes (like color). Also, there is a Form of love. Love is a feeling, a force, a relationship between a being and something else, and a bio-chemical mental state. We can then say that there is a Form of relationships like love and hate.

3) Force, as in F=ma, is a force, and therefore has a Form. There is also a Form for mass, and another one for acceleration. F=ma is a relationship describing things that already have Forms. Therefore, if we can find a Form of a relationship between things that already have a Forms, your teacher is right.

4) Justice is a relationship between an event/action, like a crime, and a particularly-satisfying result, such as a punishment. Beauty is a relationship between a set of characteristics/attributes, like big eyes, that when perceived as a whole produce pleasure in the observer. There is a Form for beauty, eye, punishment, and crime.

Your professor is correct, “F = ma” is indeed a Platonic Form. What bothers you about it is the same thing that bothers me, and the same thing that bothered Aristotle, and the same thing that seems to have bothered Plato himself later on: there is a problem with Platonic Forms.

However, your teacher did not claim that the forms are true. He claimed that “F = ma” is a Platonic Form, and he was right.

PS: Whether he was right for the right reasons or it's purely coincidental, remains unknown. Also, whether one can be right without knowing it, is a different matter. Your teacher was right when he said it, as per the meaning of "right" you had in your mind when you posted this question.

Hope this helps!

• I'm not sure I follow why a force = a form. I think the claim is that the relation itself is a Form just as Plato says harmony is one and expresses this through the equation of tones such that 1/2 as long = same tone down an octave. Feb 15, 2014 at 4:13

Well, the platonic forms don't have to be entirely separate from spatial concepts. After all, Plato explicitly referred to geometrical objects as examples of platonic forms. The form itself is an object, an that object has to be aspatial--but that doesn't mean that what the form corresponds to in the material world must be aspatial or atemporal.

Moreover, it's pretty clear that if lines can be forms, then it should be true that vectors (directed line segments) should also be forms. Now whether or not equations can be forms, seems to me a bit less clear, but I tend to suspect that they are. I think it's easy to imagine that, if Plato had known about coordinate geometry, he would admit that the 2D plane is a form--and since a circle in the 2D plane is the solution set of an equation (x-a)^2 + (y-b)^2 = r^2, then we should say that the solution set of the equation is a form. That still doesn't quite get us to the point where we say the equation itself is a form, but it's pretty close.

I'm not sure how to argue it the rest of the way; I think, for that, you might need someone more thoroughly familiar with Plato's work.

• I like the answer overall but there's two things I'm not following here. I'm not quite understanding what is meant by "the form itself is an object". What does object mean in that sentence/context? Also, I'm not sure what you mean about lines being forms. Feb 13, 2014 at 4:15
• So the form is an object itself: For instance, one form is the triangle. This is an object in Plato's ontology--it is a thing that exists, in fact, with more reality than triangular material objects. This form has no shape per se, because it is a form and therefore aspatial. Anything aspatial cannot have a shape. So the form of the triangle is not triangular, only triangular material objects are. Similarly, the form of the vector itself is aspatial, even though the material objects which are described by it have spatial dimension. Feb 13, 2014 at 4:21
• Lines are forms: If squares are forms, then it stands to reason that its sides (which are line segments) are forms. And in general, all geometric objects (or at least the natural ones, like points, lines, planes, figures) are forms. Feb 13, 2014 at 4:22
• I don't think the world "object" is a part of Plato's ontology. That's why I'm objecting to its anachronistic injection. Regarding your second point, it does not follow for Plato that if squares are forms, lines are forms. It is not the case for Plato that if humans are forms, human hands are forms. Feb 13, 2014 at 4:26
• If I may add a comment to the above comments, I think that is necessary to separate two (interrelated but different) issues: the exegetical issues regarding Plato's works and ideas, from an "adaptation" of his doctrines to modern contexts. The first is an "highly technical" discipline; the second is (for me) slightly useful. An example: "'object' as we sometimes use it implies a certain relation to the mind that thinks..."; are there "mind" and "thinking" in Plato's vocabulary ? Feb 13, 2014 at 8:35