Random Thoughts

It has been, it seems, at least two years since I have logged in to my WordPress account. However, I have not forgotten my little blog, nor my enormous ambition when I graduated from college five years ago of being a student forever and using this blog to record what I learn.

I don’t think I am ready start writing long posts again, but tonight I somehow feel compelled to log in and write about my recent thoughts.

Random Thought #1: I have been reading a book, Anatomy of an Illness by Norman Cousins. Tonight, I read the chapter entitled “Creativity and Longevity”, in which the authors writes of his encounter with Pablo Casals when he was almost 90 years old. Casals said his favorite composer is Bach but the composition most meaningful to him is Brahms’ B-flat Quartet. Apparently, it took Brahms 9 months to write the quartet, and it was completed on the same day, same month, and same year as when Casals was born… I thought that was amazing. Later tonight, I looked up the author, Norma Cousins, and found that he died a few days after I was born. Somehow, I was touched by this fact. I am part of history. One person dies, another person is born. I am alive now and it is my time.

Random Thought #2: Last night, through a search of video lectures on real analysis, I discovered the webpage of Professor Francis Edward Su, a professor of mathematics at Harvey Mudd College. I read all of his writings and found them very inspirational (especially his essay on “grace”). I was delighted to find that he is also interested in theology, and I was amazed and my heart filled with happiness when I read this quote on his webpage:

From Geoffrey A. Studdert Kennedy, The Wicket Gate [1923]:

Religion leaves a million questions unanswered and apparently unanswerable. Its purpose and object is not to make a man certain and cocksure about everything but to make him certain about those things of which he must be certain if he is to live a human life at all.

Religion does not relieve us from the duty of thought; it makes it possible for a man to begin thinking. It does not put an end to research and enquiry, it gives a basis from which real research is made possible and fruitful of results; a basis without which thinking only means wandering round in circles, and getting nowhere in the end, and research means battering at a brass door that bruises our knuckles, and does not yield by the millionth part of an inch.

This is exactly what I think. I have held this opinion ever since I became Muslim seven years ago.

Earlier this year, someone asked me, what has been the greatest psychological benefit of religion to you? I said, ever since I became a Muslim, I have stopped stressing so much over the big questions on life, such as: Why am I here? What is the purpose of life? The questions that haunted me in my teenage years. In other words, religion has relieved me of the stress and anxiety generated by the uncertainties that come from not being able to answer these questions.

However, that does not mean I have stopped thinking about these questions. Thinking about these questions was what brought me to Islam, and Islam in turn provides me with a beautiful framework to help me think even more deeply about these questions. To me, religion is a home from which I can safely explore the world. It provides me with a vantage point without which I would find the world quite a disorienting place. It gives me peace and frees up my mind to think about deeper questions in life.

It is always amazing to find out that someone felt exactly the same way I feel. That is why I love reading. I love meeting these people whom I would otherwise have never met, and learning about what they think.

It’s late now. I have to end this post. I have been corresponding with a guy in the past two months, and it is time to write him. It is nice to have someone to write to and to feel connected to another person. I don’t think anyone truly enjoys being alone. Our genes compel us to be social, even only with our minds.

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Fear breeds the very thing one is afraid of

I once read an answer on Quora that said something along the lines of “fear breeds the very thing that one is afraid of.” The example given was how germophobes are actually the ones who spread germs by peeing all over toilet seats in public restrooms because they are afraid to sit down. In this way, fear of dirty toilet seats creates dirty toilet seats.

The more I reflect on this idea, the truer I find it is. Here are a few more examples I have come up with from personal experiences and reflections:

  • I am afraid of coldness, and when I am cold, I don’t want to move because if I move, I get cold. However, the longer I don’t move, the colder I get. If I could just get over the initial coldness, moving actually warms me up. In this way, being afraid of coldness actually creates coldness for me. What I need to do to combat coldness is to confront it by moving my body against the cold air.
  • I have perfectionist tendencies. What that means is that I am afraid of failure. However, the more I am afraid of failure, the more paralyzed I become. By delaying action, I actually become more of a failure than I would have been otherwise.
  • Some people are afraid that eating too much will make them fat, so they go on strict diets. However, dieting makes people hungry, and hungry people eat a lot. At the end, they become fatter than before.

The pattern in these examples is this:

fear of an object –> avoidance of the object of fear –> growth of the object of fear

In order to break this pattern, we have to change our reaction to fear from avoidance of our imagination of the object of fear to confrontation to the reality of the object of fear.

I find that it is useful to think of what I am afraid of as an enemy. How does one defeat an enemy? The first step, of course, is “know thy enemy”. In other words, one has to gather as many facts and be as objective as possible about one’s object of fear. Often, this is enough for the fear to subside (if the enemy is weak).

Then, after learning everything about the enemy and devising tactics accordingly, one attacks the enemy by confronting it directly. This step requires a lot of courage, which is the opposite of fear. While fear creates more objects of fear, courage destroys them.

To conclude, fear breeds the very thing one is afraid of. The best weapons we have against this enemy are knowledge and courage. By investigating the enemy, we gain knowledge. This knowledge in turn fuels our courage to confront and defeat the enemy.

I learned that courage was not the absence of fear, but the triumph over it. The brave man is not he who does not feel afraid, but he who conquers that fear.” – Nelson Mandela

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[Thinking Out Loud] What to do with the feeling of emptiness

By (the Token of) Time (through the ages), Verily Man is in loss, Except such as have Faith, and do righteous deeds, and (join together) in the mutual teaching of Truth, and of Patience and Constancy. Qur’an (103:1-3)


Lately, I have been feeling an emptiness in my heart. I have been feeling kind of lost. Like, what am I doing with my life? Is this really the best way to live my life? Have I made the right decisions? What will be my future?

To be honest, I am tired of thinking so much, and I am tired of asking myself these questions. But I can’t help it. And I get stressed out all the time.

Fortunately, Qur’an offers some guidance on what to do in this situation. According to Surat al-Asr, I feel this way either because:

  1. I don’t have faith;
  2. I don’t do righteous deeds;
  3. I am not in a relationship where the other person and I can teach each other about Truth;
  4. I am not in a relationship where the other person and I can teach each other about Patience and Constancy; or,
  5. Any combination of the above.

And indeed, I am really not doing much these days in God’s eyes. I think I have faith, but my actions don’t tell the same story. I don’t really do any righteous deeds, and I am not in any relationship whatsoever. I don’t even have friends anymore.


I guess the first thing I should start working on is faith. Having faith means trusting God’s knowledge of what is best for me. And God tells us that our purpose in life is to worship Him and Him alone, so that’s what I should do.

But even though worshiping God sounds simple, it’s actually not. In Islam, worshiping God doesn’t just mean believing with your heart; it means doing what pleases God, like doing the daily prayers, respecting others, being generous, continually making progress in one’s knowledge etc.—generally, acting as if God is watching you all the time. It’s a lot of work. For example, the five daily prayers are a constant practice for self-discipline, and a physical reminder to put God as our priority in daily lives.

What’s the best way to accomplish these goals? After many years of trial and error, although I haven’t succeeded yet, I think the key lies in “living in the moment”: When working, work. When sleeping, sleep. Connect the mind completely to the body, and one may accomplish a lot and not even feel tired.

But it’s really difficult to simply live in the moment all the time. What makes human beings different, after all, is that we have extremely active and wandering minds.

One solution, I suppose, is to fill my schedule completely with work and studies. I’ve found that when I work, I actually do live in the moment. Studying is harder and requires a lot of self-discipline, but when I really force myself to concentrate, I can get in a state of flow which indicates that I have successfully synchronized my mind and body.

I can also try to “meditate” which I interpret as trying to think of nothing at all. That would probably help with my insomnia.

I like the idea of writing, too. I’ve found that when I write, I empty out my worries temporarily, which helps me with whatever I do in the rest of my day.


So here is my tentative action plan and what will hopefully follow: work/study/meditate/write a lot –> live in the moment–>accomplish goals for worshiping God–>in doing so, become a faithful Muslim–>do righteous deeds and develop relationships where the other person and I can teach each other about truth and patience–>not feel lost/empty.

Other Thoughts: Mind/Body relationship in animals, human beings, and angels

Rocks and water and other natural matter don’t have a mind (or just a proto-mind). Animals have minds, but in animals, the mind is directly attached to the body, so animals are always living in the moment, it seems. In human beings, we have a very active mind, and it’s not attached to the body; the mind travels back and forward in time while the body is stuck at one slice of time. As a result, our mind is constantly out of sync with the body.

What’s the next stage for us? If animals unconsciously synchronize bodies and minds, and human beings unconsciously misalign bodies and minds, then the wise human beings will perhaps be able to consciously synchronize bodies and minds.

And when we get really good, perhaps we’ll just return to the innocent state of animals, but with complete consciousness. That almost sounds like angels, actually. And angels are the perfect worshipers of God.

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the importance of having a perspective

I remember how confused I was as a teenager. I realized even then that adolescence was the period for me to define my character and my personality and to form my own opinions on things. But how could I decide what to like, what to believe in and what to think, when there were so many opinions in the world, and each seemed defensible in some way? Reading books and online articles on different subjects didn’t help either; my opinion on any given subject would simply shift with whatever I happened to be reading at the time.

My problem was: I didn’t have a central belief with which to compare the other beliefs or opinions I came across. Without a central belief acting as a ship (or structured protection), I was drowning in an ocean of opinions, suspending judgment indefinitely in hope of finding the truth, or at least the most correct belief. But sometimes, it’s faster to just pick an opinion first, then start exploring this ocean.

As it went, I boarded the “ship” of Islam. Finding a central belief was just the beginning of my journey though. Or, to use the analogy above—my exploration of the ocean of opinions. Equipped at last with a central belief, I found new worlds of knowledge to explore in philosophy and social sciences, all of which had not interested me in the least before.

It was an amazing experience. Philosophically, I transformed from a relativist into an absolutist (being an absolutist doesn’t mean I am dogmatic or fanatic; all it means is that I believe in “absolute” truth instead of “relative” truth). Personality-wise, I went from being an INFP (introverted, intuitive, feeling, perceiving) to an INFJ (introverted, intuitive, feeling, judging) in the Myers-Briggs personality test. That was pretty interesting as well.

I wouldn’t say which philosophical position or which personality is better, but I feel a lot happier and more comfortable now holding the view that I do. My mind doesn’t wander as much anymore, and I learn something new from everything I read.

The point is this: To see any big picture clearly, first you must have a perspective.

Here is a paragraph from a book I am reading at the moment. It describes how “perspective” works and illustrates this point perfectly:

“Perspective is a notion from the science of sight. It conceives sight as a transaction between a thing at one place in space, and another thing, a perceiver, at a different place in space, but one at which she can receive energies from the first. It is therefore essential to it that the perceiver is located in a space of stable objects on which she has a point of view. If the perceiver had no location, or was able to shift instantaneously from place to place without a speed limit, the information would not be better, but worse. It would smear and blur and fail when the displacement exceeded the speed of processing. It would stop being information at all.” p.87, Truth by Simon Blackburn (my italics)

Although I have not read any of Nietzsche’s work, I really agree with his theory of “perspectivism” (as described by Simon Blackburn). As human beings, we are naturally limited by our individual perspectives. But we don’t have to think of our necessarily having a perspective as a limit. Instead, we could think of it as a very effective tool to observe the world and learn from the world. Together, we can build a vision of the truth by sharing our perspectives with each other. Perhaps that’s why God said:

“O mankind! We created you from a single (pair) of a male and a female, and made you into nations and tribes, that you may know each other (not that you may despise (each other)). Verily the most honored of you in the sight of God is (he who is) the most righteous of you. And God has full knowledge and is well acquainted (with all things).” 49:13 Qur’an

If we get to know each other very well, we will also become familiar to our different perspectives. The more perspectives we add together, the nearer we are to the truth. Now the problem is: how do we get to know each other that well? Hmm.

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A defense of mathematical Platonism

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.


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


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]:

three worlds Roger PenroseGoing 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:

three worlds modified Roger PenroseGoing 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.

•    Physical to Platonic Mathematical: Either (a) the entire physical world is governed by the Platonic mathematical world, or (b) some physical actions are beyond the scope of mathematical control.

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

•    Mental to Physical: Either (a) the entire mental world is dependent on the physical world, or (b) some forms of mentality are not rooted in physical structures.

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

•    Platonic Mathematical to Mental: Either (a) the entire Platonic mathematical world is within the scope of reason (in principle), or (b) there are some mathematical truths that are inaccessible to reason (in principle).

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:

noumenal and phenomenal

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.


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.


[17]Ibid. 111.

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

[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.

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I don’t want to be a perfectionist anymore

I have been avoiding coming here for a long time. Not because I have nothing to write about, but because I have so much that I don’t even know where to start. But at the same time, many of my thoughts, memories, and ideas dissipate. The result is that my mind is back into to being its hyperactive and sleepless self.

I believe that there are two reasons for my predicament: (1) I read faster and process information faster than I write (like everyone else), and (2) I am a lazy perfectionist. I have many ideas that I want to write out in a perfectly logical manner (and thereby also have a clearer idea of what I really know or think).

But how could I possibly do that? It’s such a big project. So I delay indefinitely what I think is the most important part of the process of gaining knowledge: writing about what I learn.

Unfortunately, the more I delay, the harder it is to start writing again. Perhaps I could say that my mind is analogous to a room. And writing is analogous to tidying up the room. After months of not writing anything down, my thoughts can only be described as chaotic. And just like how it’s a headache to organize a messy room, the longer I don’t write, the harder it is to write when I actually want to write.

But from now on, I will refuse being a perfectionist. I don’t care what people think. I am writing only for my own benefit: I write to learn. So, I will come here everyday to write. Maybe I won’t finish a post everyday, but I WILL come here to write even for just 15 minutes a day. That’s the promise I make to myself.

That’s it for this post, I guess. Even though I do want to write a little bit about how the idea of “writing to learn” reminds me of the Socratic method. The act of writing forces me to examine my thoughts and pull them out in the form of a string of words, which act like a “logical machine” to organize my thoughts in a logical way. This action of “pulling out” is similar to the action that is performed in the Socratic method of questioning. The only difference is that I am the one who is questioning myself.

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Roger Penrose’s “three worlds and three deep mysteries” theory

A little more than a month ago, I was studying in the library and saw a rather big and conspicuous book on the shelf called The Book to Reality. I picked it up and started reading casually, but was soon absorbed in the content and fascinated by both the depth and breadth of the book. After a month, I am still reading the book. I haven’t gotten far (at all), but that’s not because the book is boring, but rather because of two reasons: 1) I am very ignorant about many topics discussed in the book (e.g. hyperbolic geometry), and 2) the book is so thought-provoking that I cannot get very far without stopping to think and write stuff in my notes, googling things etc.

For this post, I would like to go back to the first chapter and summarize and comment a little on a particularly thought-provoking section called “three worlds and three deep mysteries”.

In Penrose’s metaphysical framework, there are three forms of existence or “worlds”: the physical, the mental, and the Platonic mathematical, as illustrated in the figure below (extracted from p.18 of the book):

three worlds Roger PenroseGoing clockwise, the figure reads (the three mysteries):

  • A small part of the Platonic mathematical is relevant to the physical.
  • A small part of the physical induces the mental.
  • A small part of the mental is concerned with the Platonic.

Going counterclockwise, the figure reads (Penrose’s three prejudices):

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

However, since this view reflects some of Penrose’s prejudices, he has drawn another figure to accommodate different viewpoints:

three worlds modified Roger PenroseGoing clockwise, this figure reads the same as before (the mysteries remain).

Going counterclockwise, however, this figure now allows:

  • The possibility of mathematical truths inaccessible to reason (in principle)
  • The possibility of mentality not rooted in physical structures
  • The possibility of physical action beyond the scope of mathematical control

As indicated above, either view contains the same mysteries though; namely:

  • Why do mathematical laws apply to the physical world with such precision? Why are mathematical laws so beautiful?
  • How can some physical materials like human brains conjure up consciousness?
  • How is it that we can perceive mathematical truth? How could we grasp the actual meanings of “zero”, “one”, “two”, “three”, etc.?

The book is mostly concerned with the first mystery: the remarkable relationship between mathematics and the physical world. Regarding the second mystery, that of consciousness, Penrose believes that “there is little chance that any deep understanding of the nature of the mind can come about without our first learning much more about the very basis of physical reality.” (p.21) The third mystery is discussed briefly in the book in relation to the notion of mathematical proof.

What do I think of this theory? Personally, I prefer the second figure, the one that allows the different possibilities. This is mainly because I think that there is a very small number of mental conditions not rooted physical structures (mystical experiences), and that there is also a very small number physical actions beyond the scope of mathematical control (miracles). I am not sure if there may be some mathematical truths that are simply beyond human reason, though.

Obviously, I believe in the existence of mystical experiences and of miracles due to my religion. However, this viewpoint is also surprisingly compatible with the concept of Tai Chi, more commonly known as Yin Yang. I am not an expert, but the idea is that everything in the world exists in a binary harmony in the form of Yin and Yang (also see Qur’an 51:49). In Yin (dark/feminine), there is always a little bit of Yang (bright/masculine), and in Yang, there is always little bit of Yin. If you want to cut the circle in half, there is always going to be some black and some white, never just one color. So, perhaps it is the case that the vast majority of mental conditions are rooted in physical structures, but there is a very small number that are not. And perhaps it is the case that the vast majority of physical actions are within the scope of reason, but there is a very small number that are not.

Although I prefer the second figure, I agree completely with Penrose’s identification of the three the mysteries associated with this framework.

The relationship between mathematics and physics is indeed mysterious and endlessly fascinating. The physical world conforms to mathematical laws to such a high degree that I sometimes suspect that the whole physical world is just the “image” of the mathematical world, and that the whole Platonic mathematical world could eventually be shown to be completely intertwined with the physical—two sides of the same coin kind of thing. But of course, this is a very idealistic conjecture.

Regarding the second mystery, it’s really what the whole field of Philosophy of Mind is concerned with. Explaining consciousness is the “hard problem”! And indeed I thought very hard about this problem when I was taking Philosophy of Mind more than three years ago. The conclusion I reached was the same as Penrose’s: we still don’t have enough understanding of the physical world to figure it out. Specifically, I came to agree with the philosopher David Chalmers on this issue (Type F Monism).

The third mystery is not formulated very clearly in my mind, but I suppose that it’s related to epistemological questions like these: How do we know what we know? What exactly is the connection between the mental world and the Platonic mathematical world? Is logic everything? This is a very deep mystery, and in my mind perhaps the most difficult one to solve.

An additional thought: It seems that right now, we have the most chance solving the first mystery, and that in order to solve the second and the third mysteries, we still need many more big ideas and big discoveries. I just have a feeling that these mysteries are so deep that we probably still don’t know what we don’t know! (We have to account for the “unknown unknowns!) We can still think about these things to come up with the most plausible conjectures, though—an activity otherwise known as philosophy. Perhaps it’s futile to philosophize when we have so little knowledge, but I believe with the help of divine input (God’s Revelation) we can at least work out some theories about the “shadows” we are seeing.

Overall, I like Penrose’s metaphysical theory for its coherence and ease of understanding. At this stage of my life, I am concentrating on understanding the relationship between mathematics and the physical world, but I am really interested in the whole thing—how everything fits together. I will never reach that knowledge, because that Knowledge resides with God and the path towards God is infinite, as God is infinite. So I will just look up into that unseen mountain top and keep climbing, insha’Allah!

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