A few words about this essay: I wrote it for my Foundations of Mathematics class last semester (May 8 2015). It summarizes my thoughts regarding Reuben Hersh’s idea of mathematics as a mental model. It’s really more of a philosophy paper, and only tangentially related to mathematics. But my professor was very understanding and believed that it was more important to cultivate our lifelong relationship with math than to give us a low grade on a paper because it didn’t actually involve math. Anyway, I don’t think anyone is going to read it, but I am posting it here just as a record of my thoughts.

Note: the first section of the essay is really similar to my last blog post on Roger Penrose.

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A defense of mathematical Platonism and a Platonist theory of the metaphysical place of mathematics

__Introduction__

What is the place of mathematics in the world? Does the mathematical world exist “out there” or only in our mind? Is it discovered or invented? These are important questions in both the foundations of mathematics and the philosophy of mathematics, and they are the questions to which I attempt to address in this paper.

This paper will be divided into five sections. First, I will present Roger Penrose’s theory of the “three worlds and three mysteries” as a metaphysical framework that provides the big picture for the discussion. Then, I will summarize Reuben Hersh’s idea of mathematics as a “mental model”, as discussed in his various works. After that, I will defend mathematical Platonism by making an objection to Hersh’s theory. In the fourth section, I will elaborate on the place of “phenomenal and noumenal” mathematics in the three-world framework. Finally, I will attempt to answer the all-important question: Is mathematics discovered or invented?

First, the big picture.

__I. Roger Penrose’s “Three Worlds and Three Mysteries” Theory__

Roger Penrose is a British mathematical physicist and a mathematical Platonist who believes in the existence of a “mathematical world” independent of human minds. In the first chapter of his book, *The Road to Reality*, Penrose discusses his metaphysical theory called “three worlds and three mysteries”. Although I do not completely agree with this theory, I, as a fellow mathematical Platonist, agree with the general structure of the theory and find it to be a useful framework for organizing my own metaphysical views, as well as an appropriate springboard for our present discussion.

This theory states that there are three forms of existence or “worlds”: the physical, the mental, and the Platonic mathematical, as illustrated in the figure below[1]:

Going clockwise, starting from arrow number 1 (that connects the Platonic mathematical world to the physical world), the figure reads:

- A small part of the Platonic mathematical world is relevant to the physical world;
- A small part of the physical world induces the mental world; and
- A small part of the mental world is concerned with the Platonic mathematical world.

Going counterclockwise, starting from arrow number 3 (that connects the Platonic mathematical world to the mental world), the figure reads:

- The entire Platonic mathematical world is within the scope of reason (in principle);
- The entire mental world is dependent on the physical world; and
- The entire physical world is governed by the Platonic mathematical world.

Although not entirely uncontroversial, the statements in the clockwise reading are, generally, more accepted than the statements in the counterclockwise reading[2], which reveal Penrose’s prejudices. To accommodate those who do not agree with the statements in the counterclockwise reading of the above figure, Penrose has redrawn the figure this way:

Going clockwise, this figure reads the same as before. Going counterclockwise, however, this figure now allows:

- The possibility of mathematical truths that are inaccessible to reason (in principle);
- The possibility of mentality not rooted in physical structures; and
- The possibility of physical actions beyond the scope of mathematical control.

To summarize the Penrose’s description of the relationship between each pair of the worlds:

Pair of Worlds | Relationship (Penrose’s Description) |

1. Platonic Mathematical and Physical | • Platonic Mathematical to Physical: A small part of the Platonic mathematical world is relevant to the physical world.
• |

2. Physical and Mental | • Physical to Mental: A small part of the physical world induces the mental world.
• |

3. Mental and Platonic Mathematical | • Mental to Platonic Mathematical: A small part of the mental world is concerned with the Platonic mathematical world.
• |

These descriptions may not be entirely accurate, and they are definitely not universally accepted. However, everyone would agree that associated with these three pairs of worlds are three deep mysteries.

The first pair of worlds is associated with the mystery of the “unreasonable effectiveness of mathematics”: Why do mathematical laws apply to the physical world with such precision? And it is a pair the exploration of whose relationship is relevant to certain thinkers who argue that the world is entirely mathematical. For example, in his book *Our Mathematical Universe*, the physicist Max Tegmark argues that our universe isn’t only capable of being described by mathematics, but that our universe IS mathematics.

The second pair of worlds is associated with the mystery of consciousness: How can some physical materials like human brains conjure up consciousness? What is the nature of consciousness? Does it emerge from physical materials, or is it something fundamentally different? How can human minds have any knowledge of the physical world? These questions and others are important perennial questions in philosophy of mind and epistemology and will likely gather more attention as technologies like artificial intelligence become more mature in the future.

The third pair of worlds is associated with the mystery of mathematical knowledge: How is it that we can perceive mathematical truth? How could we grasp the actual meanings of “zero”, “one”, “two”, “three”, etc.? What exactly do mathematicians do when they “do mathematics”? Is mathematics invented or discovered?

Although the mysteries associated with the first two pairs of worlds are very deep, interesting, and important mysteries, it will be this last mystery concerning mathematical knowledge, and in particular the latter two questions raised above, that will be the main subject of this discussion.[3]

In the next section, I will summarize what Hersh thinks is the answer to the question “What exactly do mathematicians do when they “do mathematics”?

__II. Hersh’s “Mathematics as Mental Model” Theory__

Reuben Hersh is an American mathematician “best known for his writings on the nature, practice, and social impact of mathematics”.[4] In his works[5], Hersh compares three positions on the ontological nature of mathematics: Platonism, nominalism, and his own position, mathematics as mental model. In the paragraphs that follow, Platonism and nominalism will be described briefly, followed by a more extensive discussion on Hersh’s “mental model” theory.

Platonism is the position held by Roger Penrose, as mentioned in the previous section. It is the position held by most traditional mathematicians and myself. According to Platonism, mathematical objects (and their relations and structure) are real, existing “out there” in a Platonic mathematical world, and statements made about them, or mathematical statements, have definite truth values because they are about objects that are constant, eternal, and unchangeable. A mathematical Platonist, therefore, believes that mathematics is discovered and not invented. As Hersh himself describes: “According to Platonism, a mathematician is an empirical scientist like a geologist; he cannot invent anything, because it is all there already. All he can do is discover.”[6] Famous mathematical Platonists include René Thom and Kurt Gödel, who says “Despite their remoteness from sense experience, we do have something like a perception also of [mathematical objects]… I don’t see any reason why we should have less confidence in…mathematical intuition, than in sense perception… They, too may represent an aspect of objective reality.”[7]

Mathematical nominalism (hereafter “nominalism”) is the position according to which either mathematical objects do not exist at all, or they do not exist as abstract objects in the Platonist sense. For the sake of discussion, I will call the former view “strong nominalism” and the latter “weak nominalism”. Weak nominalism is more popular and is the one held by William of Ockham, the “prince of nominalists”. He wrote, as quoted by Hersh: “No thing outside the mind is universal… It is just as great an impossibility that some thing outside the mind be in any way universal… as it is an impossibility that a man be an ass.” It could be said then that weak nominalists believe that mathematical objects could exist, at most, only in the mind.

“Mathematics as mental model” is the name I give to Hersh’s own position. It is the position with which Hersh attempts to remedy the inability of either Platonism or nominalism in describing, in his opinion, what mathematics actually feels like in “living education in mathematics and… living research in mathematics”.[8] Hersh thinks that mathematical objects do not exist as abstract objects in some Platonic mathematical world, but also does not agree with the strong nominalist that they do not exist at all. Rather, his position is closer (and in fact, I would argue, essentially identical[9]) to that of a weak nominalist: mathematical objects exist only in the mind. The only two differences between the two positions, from what I can tell, are: (1) Hersh specifically points out that since mathematical objects are mental objects, in some form or realization, they are also physically present in our brains, so they do “exist” in space-time, and (2) He doesn’t just say that mathematical objects exist in the mind, but also describes *how* they exist in the mind, i.e. as mental models.

In explaining what exactly he means by a mental model, Hersh wrote:

I use the expression “mental model” for the internal entity in the mind of anyone, including a mathematician, any entity, object, or process that one may think about, concentrate on, study by inner thought. A mathematical concept is a collection of mental models that are “mutually congruent,” fit together. The concept of “triangle”, for example, is a shared, public, inter-subjective entity. Each of us who “understands” the word “triangle” has his/her own internal entity, available for contemplation or mental manipulation. That inner, private mental entity corresponding to the shared concept is what I mean by our “mental model.” Under the pressure of a strong desire or need to solve a specific problem, we assemble a mental model which the mind-brain can manipulate or analyze.[10]

In this way, “established mathematics” can be described as a collection of mental models that exist individually and collectively in the minds of experts in different fields of mathematics, and can be said as having a “mental-social reality”, as opposed to an “abstract reality” of Platonism or a “lack of reality” of strong nominalism.

This is a useful concept, for by referring to it, we can now describe what exactly goes on when a mathematician proves a theorem or constructs something new, or when a student grasps a mathematical concept. When doing poofs, “The mathematician leads the learner to observe and manipulate his/her own mental models, enabling her/him to “see—to apprehend directly by observing his/her own mental model—the claimed attribute or property of the mental model in question.”[11] When constructing something new, the researcher inspects manipulates her own mental models of mathematical concepts. And when learning a new concept, “The successful student reconstructs what is already known by others to acquire his own mental model of a math concept.”[12]

These explanations using the concept of mental models certainly make a lot of sense. But what evidence does Hersh provide for the existence of mental models? The evidence he provides consists of mathematicians’ accounts of what they feel like they are doing when they do mathematics. Almost universally, mathematicians describe mathematical activity as a tangible experience involving something analogous to direct perception: “I have myself always thought of a mathematician as in the first instance an observer, a man who gazes at a distant range of mountains and notes down his observations.” (G.H. Hardy) “I confess I have great difficulty distinguishing my activity from that of an entomologist.” (Schutzenberger) “Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion… You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated.” (Andrew Wiles) “A mathematician lives in an intellectual landscape of definitions, methods, and results, and has greater or lesser knowledge of this landscape. With this knowledge, new mathematics is produced, and this invention changes more or less significantly the existing landscape of mathematics.” (David Ruelle and Alain Connes)[13]

Indeed, all these accounts do support the existence of mental models as described by Hersh. But note that the nature of these accounts is very similar to that of Gödel’s account of his feelings about mathematical objects, quoted earlier as an aid for describing what Platonists believe. And with this observation, we have come to a very important point: Platonists do not deny the existence of mental models. What Platonists do deny is that mathematics is nothing *but* mental models.

But if not, what else can it be? How should we characterize the reality of mathematics? This is the point on which I will elaborate in the next two sections.

__III. Mental Model as Evidence for the Existence of an Actual Mathematical World__

In comparing his view to the Platonist view, Hersh wrote: “Mathematical concepts are real entities, not fictions. Platonism mistakenly locates these entities ‘out there,’ in an external unspecified realm of non-human, non-physical reality. But they are right here, in our own individual minds, shared also with many other individual minds.”[14]

As a Platonist, I do not disagree with the statements made in the first and third sentences of the quote. Indeed, mathematical concepts are real, and exist in our minds, and I even agree whole-heartedly with Hersh that they exist in our minds in the form of mental models. What I do disagree with is Hersh’s conclusion that the ontological nature of “mathematics” is mental.

My objection to Hersh’s conclusion may be introduced with this line of questioning: If the mathematical world is nothing but a mental model (or a collection of mental models), *then what is it a model of*? Why do we even have this mental model? How could mathematicians be in unanimous agreement about a mathematical “mental model” like a triangle and its properties? Wouldn’t it make more sense for their mental models to correspond to actual triangles with those properties, just as it would make sense for a few witnesses’ perceptions of seeing a man with a hat to correspond to an actual man with a hat?

I argue that the very existence of the mental model itself is evidence for an actual mathematical world “out there”, in the same way that the existence of one’s mental model of the physical world is evidence for an actual physical world “out there” (if one does not accept Cartesian skepticism). The corollary of this argument is that one could only disbelieve in the existence of a mathematical world to the extent that one disbelieves in the existence of a physical world.

For the reader who does not accept the idea that we have a “mental model of the physical world”, I would recommend reading the book *Making Up the Mind: How the Brain Creates our Mental World *by Chris Frith, a neuropsychologist at the University College London. We usually think that there is nothing doubtful about our direct perception of the physical world. Sure, we get tricked by optical illusions, and sometimes we mishear people or misread words. But in general, what we are perceiving IS the physical world itself, right? This is the exact premise that is being overturned in this book, which contains numerous examples of how our brains ignore, add, and hide information, lie, distort reality, even mix up the senses. According to Frith, what we think we perceive directly as the physical world is actually not the physical world, but the *mental model of the physical world* that our brains create for ourselves (“My perception is not of the world, but of my brain’s model of the world.”[15]). In describing this mental model, Frith wrote: “What I perceive are not the crude and ambiguous cues that impinge from the outside world onto my eyes and my ears and my fingers. I perceive something much richer – a picture that combines all these crude signals with a wealth of past experience. My perception is a prediction of what ought to be out there in the world. And this prediction is constantly tested by action.”[16] The title of Chapter 5 of the book sums it up succinctly: “Our perception of the world is a fantasy that coincides with reality.”[17]

But even though all we perceive is our mental model of the physical world, only someone crazy would seriously consider the possibility that there isn’t a *real* physical world existing “out there”. Surely, a sane person would never say that everything he perceives as the physical world is just all in his head!

The same reasoning can be applied to the relationship between Hersh’s (mathematical) mental model and the mathematical world. Mathematicians past and present have reported “perceptions” (for the lack of a better word; but maybe we could say “apprehensions”) of their mental models with remarkable coherence and consistency, as Hersh himself noted. It would be very unlikely indeed for their mental models to not correspond to an actual mathematical world existing “out there”. And I think it would not be overly critical for me to describe someone who does not believe in the existence of an actual mathematical world as crazy.

In fact, I would even argue that, if anything, we should put more faith in the existence of the mathematical world than in that of the physical world, for the degree of coherence, consistency, collective consensus and individual feelings of certainty among mathematicians about their mathematical mental models is certainly higher than that among people about their mental models of the physical world.

For the reader who objects to the above argument by citing that “argument by analogy” is logically fallacious, here is my response: Such an argument is only invalid when it relies on a property that the things involved in the analogy (comparators) do not really share. If the argument relies on a property that the comparators *do* share, then the argument is valid.[18]

The comparators in my argument are “the mathematical world mental model” and “the physical world mental model”, and it’s true that the two have some very different properties: the inputs used to construct each are different (concepts vs. sense data), the associated “perceptive” organs of each are different (perception vs. understanding/apprehension), and the qualities of our “perceptions” of them are different, etc. However, the two comparators do share one fundamental property: they are both established mental models that exist congruently in different persons’ minds. And *this* is the property on which relies my argument, which, simply stated, would be: “The existence of a mental model implies the existence of an object being modeled.” Since my argument relies on a property that the two comparators do share, it is valid.

__IV. The Phenomenal and the Noumenal Mathematical World__

It should now be clear that when we say “mathematics” or “the mathematical world” (for the purpose of this paper, they have the same meaning), we are often referring only to our mental model of the mathematical world, and not the *real* mathematical world. But if the ontological nature of mathematics is not mental, then what is it?

To attempt to provide an answer to this difficult question, I will enlist the help of Plato and Kant.

In Book Seven of *The Republic*, Plato describes the nature of human knowledge with what has come to be known as the Allegory of the Cave, the summary of which I quote below (from Wikipedia):

Plato has Socrates describe a gathering of people who have lived chained to the wall of a cave all of their lives, facing a blank wall. The people watch shadows projected on the wall by things passing in front of a fire behind them, and begin to designate names to these shadows. The shadows are as close as the prisoners get to viewing reality. He then explains how the philosopher is like a prisoner who is freed from the cave and comes to understand that the shadows on the wall do not make up reality at all, as he can perceive the true form of reality rather than the mere shadows seen by the prisoners.[19]

In the context of our discussion, mathematicians represent prisoners of their minds. Just as how our perception of the physical world is always going to reflect the tool set we have for perceiving, our apprehension of the mathematical world is always going to reflect the tool set we have for understanding, which is the mind. Unfortunately, when the cave represents the mental world, even the philosopher cannot be freed from knowing no more than the shadowy reality.

This epistemologically pessimistic view is what has been given its canonical expression by the German philosopher Immanuel Kant: We will only have knowledge of the *phenomenal* world and never the *noumenal* world. In Kantian philosophy, phenomena are presentations of how things appear to us, and noumena are “things-in-themselves”.

Applying this concept to the present discussion, our mental model of mathematics is the phenomenal aspect of the mathematical world, and the *real* mathematical world is the noumenal mathematical world.[20] If we combine this idea with the three-world model of reality introduced in section I, we will get the following diagram:

This diagram implies that our knowledge of mathematics will forever be in the form of our mental model of the noumenal or Platonic mathematical world (and this model would be called the phenomenal mathematical world using my adopted Kantian terminology). While this means that we will never directly know the mathematical reality, this doesn’t mean that we cannot continue to refine our mental model so that it is an ever better reflection of that reality.[21] The same can be said about the reality of the physical world.

So what is the ontological nature of mathematics? The short answer is: We will never know. The long answer is: We will never know, but we will have a chance to know the ontological nature of our “mental model of mathematics”, and perhaps what *appears* to us as the ontological nature of mathematics.

At this point, Hersh would criticize that I am mythologizing and spiritualizing the mathematical world as something superhuman and transcendental, and would accuse me, as he accused another mathematical Platonist, of making a claim that is “…neither verifiable nor refutable. Anyone is free to believe it or not. And it is incompatible with ordinary scientific discourse, which long ago rejected the dualism of separate incomparable ‘Substances’ called ‘Spirit’ and ‘Matter’.”[22]

Here is my response: (1) The claim that there exists a Platonic mathematical world can be verified with the evidence that is the existence of a congruent mental model within the minds of past and present mathematicians, as has been done in section III of this paper. (2) The claim may not be refutable, but then so is the claim that a physical world exists. (3) Sure, anyone is free to believe it or not, but someone who chooses not to believe in the existence of a Platonic mathematical world would also choose not to believe in the existence of a physical world, if he is logically consistent. (4) Scientific discourse, by nature, can only involve “Matter”, otherwise it would not be called “scientific”. In philosophical discourse, the goal of which is to uncover the total metaphysical reality and not only the physical reality (as is the case for science), dualism is still an important position and has definitely not been “long ago rejected”.

__V. Is Mathematics Invented or Discovered?__

Having established the place of mathematics in our metaphysical model, we can now provide an answer to one of the questions posed in the PBS Program “The Great Math Mystery”: Is mathematics invented or discovered?[23]

A strong nominalist would answer that mathematics is invented, and a weak nominalist like Hersh would agree. A Platonist, however, would not hesitate a second to say that mathematics is discovered and not invented, just as a sane person would not hesitate to say that he did not “invent” the physical world he sees. And of course, the Platonist position is what has been defended so far in this paper.

But with the mental-model-based diagram of the phenomenal and the noumenal mathematical world presented in the previous section, we are now in the position to provide a more subtle Platonist response to this question: Mathematics is *both* discovered and invented in the sense that it is discovered, *in the form of*, or *through*, invention; more exactly, the invention of *mathematical concepts*.[24]

To clarify, I will go back to the physical world analogy. As explained before, when I see a tree, what really happens is that my mental model of the physical world is telling me that I see a tree. So, am I “discovering” the fact that there is a tree? Yes, but this discovery is made, and is only possible, through invented concepts about the physical world such as “tree”, “space”, “time”, etc. But even though the concepts “tree”, “space”, and “time” do actually correspond to some aspects of the noumenal physical world, we cannot say anything about the nature of existence of such aspects since we do not and will never have direct knowledge the noumenal physical world.[25]

Suppose I then make an additional discovery, say that the tree is just one of many trees in a forest; when that happens, my mental model of the physical world is modified so that I perceive this bigger and more accurate picture of a tree in a forest, the process of such modification would inevitably involve other invented concepts such as “forest”, “big”, “small”, “one of many”, etc. Even though these concepts all correspond to certain aspects of the noumenal physical world, these concepts are just that—concepts. They help me make sense of and navigate the physical world, but they are not the real things. In this way, I am discovering the physical world through the invention of physical concepts.

Similarly, when mathematicians discover the natural numbers, their discovery is made, and is only possible, through invented concepts about the mathematical world such as 1, 2, 3, 4, 5, etc. These concepts in the mathematicians’ mental models correspond to certain aspects of the noumenal mathematical world, but we cannot say anything about the nature of existence of such aspects since we do not and will never have direct knowledge of the noumenal mathematical world.

Suppose a mathematician then discovers negative numbers; when that happens, her mental model of the mathematical world is modified so that she could perceive or apprehend the bigger and more accurate picture of numbers in the mathematical world, and the process of such modification would involve additional invented concepts such as “negative”, “positive” etc. But again, though these concepts do correspond to certain aspects of the noumenal mathematical world, they are not the real things; they are just concepts invented to help us make sense of the mathematical world. In this way, we could say that mathematicians are discovering the mathematical world through the invention of mathematical concepts.

__Conclusion__

With the help of Roger Penrose’s three-world framework, Reuben Hersh’s “mental model” concept, the self-evident fact that “The existence of a mental model implies the existence of an object being modeled,” and Kant’s concepts of phenomena and noumena, we have obtained a better understanding of the metaphysical place of mathematics, which has allowed us to provide an alternative Platonist response to one of the questions associated with the mystery of mathematical knowledge: Is mathematics invented or discovered? We have answered that it is both.

Of course, one could still reject the existence of a Platonic mathematical world, just as one could still say that he is really a brain in a vat, or is being controlled by an evil demon, or lives in a virtual world. But, to quote Connes: “My position cannot change… it’s humility finally that forces me to admit that the mathematical world exists independently of the manner in which we apprehend it.”[26]

Needless to say, there are still a lot of mysteries to be solved regarding the relationship between the mental world and the mathematical world. For example: How vast is the mathematical world? How good can our mental model get? Etc. Almost certainly, in order to answer these questions, we cannot ignore the relationship between each of these two worlds with the physical world. Ultimately, though, these “worlds” are not separate at all, but merely concepts in our minds that reflect some aspects of universe. Understanding the true reality would thus require not only solving all the mysteries associated with each pair of worlds, but also the inherent impossibility for body-possessors of grasping, somehow all at once, the necessarily singular noun of the Truth.

[1]Penrose, Roger. “The Roots of Science.” *The Road to Reality: A Complete Guide to the Laws of the Universe*. New York: A.A. Knopf, 2005. 18. Print.

[2]Here are some examples of people who would not disagree with some of the statements made in the clockwise reading: someone who believes that mathematics emerges from the physical world, or even that the physical world is equivalent to mathematics, would disagree that only “a small part of the Platonic mathematical world is relevant to the physical world.” And a panpsychist would disagree that only “a small part of the physical world induces the mental world,” for they believe that mind is a fundamental feature of all things.

[3]A “bonus” mystery that is, in a way, related to all three pairs of worlds is the mystery of mathematical beauty: Why is mathematical so beautiful? This will undoubtedly be a good subject for another discussion.

[4]“Reuben Hersh.” *Wikipedia*. Wikimedia Foundation. Web. 08 May 2015.

[5]The content that is summarized in this section comes from three pieces of writing by Reuben Hersh: (1) the paper “How Mathematicians Convince Each Other or ‘The Kingdom of Math is Within You’” from his book, *Experiencing Mathematics*, (2) his book review review of David Tall’s book, *How Humans Learn to Think Mathematically *from Vol. 122 of *The American Mathematical Monthly*, and (3) his 1981 book cowritten with Philip J. Davis, *The Mathematical Experience*.

[6]Davis, Philip J., and Reuben Hersh. “From Certainty to Fallibility.” *The Mathematical Experience*. Boston: Birkhäuser, 1981. 318. Print.

[7]Ibid. 318-319.

[8]Hersh, Reuben. Rev. of *How Humans Learn to Think Mathematically. Exploring the Three Worlds of Mathematics*, by David Tall. The American Mathematical Monthly Vol. 122, No. 3 March 2015: 295. Print.

[9]This will be discussed in section III of the paper.

[10]Hersh, Reuben. “How Mathematicians Convince Each Other or “The Kingdom of Math Is Within You”” *Experiencing Mathematics: What Do We Do, When We Do Mathematics?* 98-99. Print.

[11]Ibid. 91.

[12]Hersh. Rev. of book by David Tall. 295.

[13]Hersh. 90, 91, 92, 93.

[14]Hersh. 90-91.

[15]Frith, Christopher D. “Our Perception of the World Is a Fantasy That Coincides with Reality.” *Making up the Mind: How the Brain Creates Our Mental World*. Malden, MA: Blackwell Pub., 2007. 132. Print.

[16]Ibid.

[17]Ibid. 111.

[18]“Argument By Analogy.” N.p., n.d. Web. 08 May 2015. <http://c2.com/cgi/wiki?ArgumentByAnalogy>

[19]“Allegory of the Cave.” *Wikipedia*. Wikimedia Foundation. Web. 08 May 2015.

[20]Kant himself thought of mathematical knowledge as *synthetic a priori* knowledge, but here, I will not write about Kant’s theory about mathematics. Rather, I will only take advantage of his concept of the phenomenal and the noumenal worlds to build on our three-world model of reality.

[21]An interesting question to ask here would be: How good can it get?

[22]Hersh. 97.

[23]The other question posed in the program is the question associated with the first pair of worlds presented in section I, namely: “Why do mathematical laws apply to the physical world with such precision?” Unfortunately, this is beyond the scope of this paper.

[24]This is very close to Mario Livio’s view as expressed on the program.

[25]This does not mean that we cannot have knowledge of the ontological nature of our “mental model of the physical world” and what *appears* to us as the ontological nature of the physical world. Indeed, such knowledge is the goal of every physicist.

[26]Hersh. 96.