In this blog post I will explore what realism entails, whether or not it is a valid claim in any domain. At the moment I have in mind four types of realism: Platonic, mathematical, scientific, and artistic. Platonic realism concerns where Plato’s forms or ideas resided. The forms were considered the true reality, where those things in the everyday world were only considered faint copies. Mathematical realism is more or less a form of Platonic realism. The main difference is that only mathematical truths reside there. Scientific realism claims that what science has shown to exists does indeed exists, such as atoms. Finally, artistic realism concerns artwork which comes as close to the everyday world as the artist can make it, showing as much detail as possible. I will take each of these in turn and speak on whether or not I see them as holding some aspect of truth to them.

I will start with Platonic realism. This form of realism asks us to see the everyday world around us as an imperfect reflection of the true forms that exist in some type of heaven (i.e. a non-physical other world). These forms are also sometimes referred to as ideas. I guess this might be because they are the ideals as in being perfect specimens. This is why nothing in our everyday world is perfect according to this type of realism.

So, why did Plato take this slant. One reason is try as you will you cannot draw a line because it has no width, only length. The same can be said for any other geometrical figure. There is no way to draw the perfect anything, let alone a geometrical figure. So, the idea of say a circle needs to have a perfect form. And, all these forms reside in the wherever. I mention geometry here because it was said that “let no one ignorant of geometry enter here” was placed above the entrance to his academy—at least this what is usually said.

But, these are not the only things that reside the Platonic realm. There are also things like virtues. These are concepts such as justice, happiness, and honesty. The highest of all of these is “the good,” and this not geometry may have been the ultimate reason for Plato to introduce us to his perfect world of forms. We may not be able to determine the virtues’ exact nature in the world of forms because Socrates was famous in Plato’s dialogues for asking about these things and never being able pin down their perfect nature. There always seemed to be a hitch to whatever definition Socrates and his interlocutors arrived at. Even if they could have been successful, whatever can be observed in our everyday world is a pale comparison to what resides in the world of forms, so no one would be able to exhibit any type of perfect virtue. But, you can rest assured that they exist.

In addition, the perfect instances of everyday objects also reside in this Platonic world of forms. You could find the perfect table, chair, bed, or house there, encompassing the ideal of each object. These objects we see in our everyday world pale in comparison once again with those existing in the world of forms. You would also need to include the living world, so you will also find the ideal cat, mouse, elephant, or any other creature there.

So, how did Plato know that there was a realm of true forms where reality really resided? He did not, but he did make up a story that I will not relate accept to give you the title, and where it can be found. I have become bored with it after having heard it related in so many books. The name of the story is the “Allegory of the Cave” in his Republic. It is a big book, so here is a link to just the story – https://web.stanford.edu/class/ihum40/cave.pdf. Regardless, the story is just—a story and nothing more; it does not prove anything. Although I do not know if it was intended to be a proof or not, it could be seen as a way of illustrating a better world than our own.

One attraction to the theory of forms involves the question of universals. In other words how do we form general concepts. According to this theory you find them in the Platonic realm—plain and simple. Aristotle did not see universals as presenting much of a problem, so he argued against the need for the Platonic solution. My own sense is all we are doing with general concepts is giving a name for a class of individuals (even abstract ones). All cats makes up the class of cats, all houses makes up the class of houses, all triangles make up the class of triangles, and so on and so forth. In the late middle ages William of Ockham supported this general view on universals, although maybe not in this form (the concept of classes was not formed back then), and it became known as nominalism. And, its opposite is realism of which Plato’s theory is just one. Another oppositional view to both of these is idealism—all reality is in the mind. But, I am not going to criticize idealism in this post (except – see below); although, there is plenty to criticize. I think the “class” thing is probably the best way to deal with universals, but again I am not discussing this here.

Here are my reasons for not believing in the theory of forms with true reality dwelling elsewhere than in our everyday world. My main issue with this theory is where is this reality supposed to be. Obviously, not in Plato’s cave. Even if forms did reside in another plane than the everyday world, how could we gain access to them. That to me is a big problem. What the universe is made of and how it works is the domain of science. And, science, at the very least in practice, takes a naturalist view of the universe. Science, almost by definition, cannot investigate the Platonic realm of true forms, hence why we have no access to it. Another thing is, is there really a perfect thing anything?* What about geometric figures? Should not there be a perfect triangle? No. A triangle is a definition, it has no reality in itself; it only shows what is necessary for something to be called a triangle.

Well, I think I have more or less dealt with Platonic realism. So, what about mathematical realism. People, mathematicians mainly, who believe in mathematical realism believe that there is a realm where all mathematical truths reside. Some also believe that numbers themselves actually reside there. But, I have heard it described both ways. This realism is very similar to Platonic realism, except the mathematical realist is only concerned with mathematical truths or objects, not with regular objects (e.g. houses, cats, mice, etc.) or virtues (e.g. justice, honesty, courage, etc.); although, mathematicians find great virtue in proof, as well they should.

Plato was a mathematical realist, as were Gottlob Frege, Kurt Gödel, and G. H. Hardy. Frege was the first to attempt to put mathematics into a logical form. He was not successful. First, Bertrand Russell pointed out a contradiction within his system, which Russell himself attempted to correct, but Gödel put a nail in the coffin of logicalism with his theorems (consistency and incompleteness). However, Gödel still was a realist when it came to mathematics at least. Hardy was a superstar of number theory. He consider even the hint of applied mathematics as a sin against real mathematics. He felt his research into pure numbers had no practical applications (you can argue that he was mistaken). So, where else would he find mathematics other than in a realm of purity.

Mathematical realism often comes down to whether mathematics is discovered or invented. For the discovery camp they feel that there must be something out there to be discovered, hence their liking for another realm where there is waiting to be discovered mathematical truths or the properties of numbers. Under discovery if one is to go about discovering mathematical truths, like a scientist discovering atoms (more on atoms below) one must find them somewhere. Might the mathematical realist suffer from a touch of science envy? What if mathematics is to be found in the mind/brain?^ Can this be construed as discovering it? I am not quite sure about this.

I am not sure because this is where mathematical truths and objects are thought to be by some of those who claim that mathematics is invented. This is mathematics with its truths and its objects are produced by the mind. So, it might be better to say that mathematics is created rather than invented, but I think invented is the more popular of the two terms. One question asked is if mathematics resides in the mind can it be objective (one bonus of the realist position)? It becomes objective, however, when mathematical proofs have been accepted by the mathematical community. In other words you could say it becomes part of the public domain, and that is where mathematics’ objectivity resides. After all, an item in our everyday world could be considered as part of the public domain. A somewhat halfway position is that it becomes intersubjective when it is shared between minds. Some claim the objectivity of our perceptions of the universe are really at best intersubjective anyway.

So, mathematicians have a choice—either mathematical truths are real in the sense that they exist whether or not any mathematician has discovered them or not, or mathematical truths are invented (or created). Is there no other choice? I think there may just be; it is possible to construe mathematics as being constructed.† I do not mean in the constructivists sense. The constructivists only accept mathematical proofs that actually construct the mathematical objects of interests, hence there is no relying on things like infinite numbers, limits, or non-finitary proofs. I mean constructed in the sense that the mathematician constructs a proof—any proof. This I think conforms to it being discovered or invented because either way the proofs of mathematics get constructed.

Does construction then solved this issue between discoverers and inventors? Probably not to the satisfaction of many mathematicians. But, more importantly does this show that mathematical realism is a not a valid ontological¹ position? At the very least it calls it in to question.

So, what is my verdict on mathematical realism? It goes the way of Platonic realism. I see know way of knowing that mathematical realism is true, so seeing how the onus is on those who posit entities to prove that they exist (it is the same problem for those who claim that god exists) and having good reasons for a naturalist’s position, the likelihood of this type of realism to be correct is slim enough to be ignored. Anyway, I think mathematicians can get on with their work without a separate mathematical realm (they would have to if it did not exist). And, some claim that mathematics is what mathematicians do.

Next up: scientific realism. So why do people claim this type of realism, and how does it differ from the first two? For the why, I think that this type of realism survives Ockham’s Razor, so this position becomes the best explanation for the existence of things.² In other words it is simpler to claim only one world where things exist. Just like it is simpler to leave god out of any explanation. What does this simplification prove? Nothing, but it does indicate where the best explanation is likely to stand—it is the most likely to be true. How it differs is that this type of realism is found in the physical universe itself.

So, what kind of entities, without any unnecessary ones, does science posit to exist? But, first what about our everyday perceptions? It is easier to believe that our brains create these perceptions base on the information coming into our sensory organs than any other explanation. Science has done a decent job of explaining what is going on in our nervous systems when we perceive things. And, before brain science was on the scene, autopsies show the existence of our internal organs, of which the brain is one.

Then, with microscopes it was possible to see things that we could not with the naked eye. Both those things we see with our own eyes and those things we see with the help of instruments of magnification allow us to see things we believe to exist without much theoretical knowledge. There are telescopes too, but with them we look at big things with significantly more theoretical knowledge. So, what about those things that are deduced as existing. Atoms once fit this bill. It helped to explain chemical reactions, though at the time no one had seen an atom or its actions upon other entities. And, it seems that atoms were necessary to explain the phenomena of heat and entropy (the tendency in a close system for disorder to grow). Well, all fine a good, but what about direct observations of these posited atoms? Einstein put the existence of atoms of on firmer ground. He explained Brownian motion (the movement of ink in a solution) by the action of atoms jiggling the ink around. Close right?

Yes, but science could do better with the help of quantum physics, which Einstein help to produce in the beginning by showing that light could come in packets—a quantum of light—building on the work of Max Planck, who showed that energy only showed itself at discrete levels. Earlier, from other experiments, it was deduce that atoms were not the basic entities that they were thought to be. It turns out they had a nucleus, which were surround by elections (originally thought to be in discrete orbits). Then, experiments start to crash atoms and their constituent particles together in accelerators. With the help of cloud chambers, and later bubble chambers, experimenters could see tracks laid down by these particles. Each particle created its own unique track. With higher and higher energy levels the protons and neutrons that made up the nucleus of an atom were shown to contain particles of their own. These were the quarks.

So, science posits all these particles, and there were plenty of them (about 200) until Murray Gell-Mann and others tame this “zoo of particles” as it was called. There were now just six quarks which make up what are called baryons, and three different singular particles, like the electrons, which are called leptons, making up what are known as fundamental particles. Anyway, even though this seems like a lot of entities being multiplied, it was the simplest system (now called, along with the four fundamental forces and the particles that mediate them, the standard model) to be devised that would cover everything that was currently known. To posit other entities that do not make up the physical universe with its forces is to multiply more than is necessary.‡

Okay, but has anybody actually seen an atom, if that be necessary to prove its existence (which it does not)? Well, they are now able to trap such things as atoms and electrons and have visualized them in color generated images. This is as close as we have come, at least visually.

As you might have guess I see scientific realism as a valid way to look at our universe. There is probably plenty to quibble with here, but I challenge anybody to have a simpler explanation of the way things are and that they exist. I discount idealism, where everything is in the mind, as it is posited without proof. With idealism you have to decide somehow, what mind contains what, and what mind might contain it all. The usual answer is that it is the mind of god. Well, if you posit this mind, you have just done an unnecessary multiplication of entities.

Okay, now for the last version of realism I want to discuss—artistic realism. So, what is this type of realism? It is the attempt to create a piece of art (e.g. statue, painting, etching) that looks as close to what we perceive it as in life. It is opposed to much romanticized artwork. Some examples of this type of artwork is British landscape painting and impressionism. And, then you have modern art and totally abstract art.

Here is a mosaic of various examples of realism in art:

Leonardo da Vinci – Lady with an Ermine

[I will remove any of these images if they are found to infringe on someone’s copyright]

Going clockwise they are: The School of Athens by Raphael, Lady with an Ermine by Leonardo Da Vinci, an example of greek statuary, The Thinker by Rodin, Statue of David by Michelangelo, and Daniel in the Lion’s Den by Peter Paul Rubens. My favorite part of The School of Athens is Diogenes reading a book in a beam of sunlight in the middle of the picture. And, Daniel in the Lion’s Den has always been one of my favorites.

While other forms of art have some interest to me, I am very partial to realism in art. I have been an abstract artist** myself, so obviously I have an appreciation of other forms of art. My least favorite form is impressionism. All art I think is a form of self-expression. Something of the mind of the artist is expressed in a piece of artwork. Other than this there are lots of different views on what art is. One definition is that art is the expression of beauty. But, beauty is even harder to define than art.

I would like to mention surrealism before I leave this discussion on artistic realism. It is very similar to regular realism in art. The pictures have realistic looking objects, but are skewed in some manner than you would see them as in reality. My two favorite surrealistic artists are Salvador Dali and M. C. Escher. A lot of Escher’s work does not involve realism, some only relate to different aspects of symmetry with non-realistic looking objects. Surrealism incorporates the blatant violation of reality, but other than that the attention to detail is the same as realism.

Here is another mosaic of images from the surrealists camp (the top two are by Dali, the side and bottom are by Escher):

[I will remove any of these images if they are found to infringe on someone’s copyright]

I have mentioned beauty above, and I would like to end on mathematical beauty. While it is not directly related to mathematical realism, I feel that most mathematicians have a sense of mathematical beauty. This maybe more so for the mathematical realist, or so I imagine. Does it give anymore support to it? Not really, but I am sure that there are arguments for it that include beauty as a component.

To end I will give my verdict on the various types of realism I have discussed in this post:

Platonic realism – false. Mathematical realism – false. Scientific realism – true. Artistic realism – neither.

¹ Dictionary.com defines ontological as: “of or relating to ontology, the branch of metaphysics that studies the nature of existence or being as such.”

² http://math.ucr.edu/home/baez/physics/General/occam.html has a good summation of this princple. However, its common formulation as “entities should not be multiplied beyond necessity” is said by both wikipedia and the Stanford Encyclopedia of Philosophy not to be Ockham’s exact formulation in any of the available texts. He did write: “Plurality must never be posited without necessity.” I feel that this is just nitpicking. Do not they mean basically the same thing? I suppose that “plurality” could be broader than just “entities.”

* Nothing known is perfect except the philosopher’s god, and this god does not exist.

^ When it comes to philosophy of mind I am a materialist. To see more on my materialist’s views look at my post – Why Are People Afraid of Their Brain?

† See Can Mathematics Be Constructed? for more of an elaboration on this view.

‡ I have described everything here as particles and forces, but the reality is a little bit different. There is now quantum field theory, which replaces quantum mechanics, and this theory posits fields, where the particles are excitations in these fields. This theory helps to better understand the universe and the way it does its thang. Sorry that I cannot do a better job in describing quantum field theory, not that I really gave a good description of the quantum mechanical picture, but I am more familiar with the latter rather than the former.

** In the past I created fractal artwork, which I suppose is an abstract art form (in another post I will explore whether or not it is art at all), even if it is based entirely in mathematics. However, computer images (any image really) can never show a complete fractal because they cannot be created to show the infinite level of self-similarity of any fractal. Self-similarity is one of the hallmarks of a fractal. All fractals look the same at whatever scale of magnification you produce them. On my home page you will find one of my images. Here is another:

The self-similarity of the Mandelbrot set is seen upon the magnification of the border, where you find buds at every level in between other buds.