In this post I talked about a book that I’d be publishing soon. Well, I decided not to publish it yet, and instead I’m putting up the first edition for free here. There’s more explanation on that page, so I’ll spare the reader as much duplicate reading as possible by ending this post here.
Hey all, again no math, sorry, although I hope you do/did enjoy the most recent lesson on dihedral groups. As I mentioned in my last blog post (“My Absence“), I’m currently in the process of finishing up lecture notes for an intensive three-week long high school course. Seeing as the course starts tomorrow, I’m just about done with them. Upon polishing them up, though, I got an idea.
I’ve received emails from readers who have read through the entire site and who want to keep reading and exploring math. This is absolutely fantastic, and you have no idea how happy emails like that make me. The problem is, however, that I have very few ideas for “what comes next” (i.e., what comes after the site (there’s still lots of content that I will be providing here in the future)), because quite honestly the only thing that comes next is a hardcore college math textbook. These might be a little daunting for someone who’s fresh off my site, but there simply is nothing to fill this void (that I know about, at least).
This brings me to my idea. Instead of just posting my lecture notes online, what I’ve done is turn them into a full-fledged book. It’s well over a hundred pages of math (and some physics), some of which we’ve seen here already, and the vast majority of which will comprise the next few hundred lessons on the site. The book will, of course, move much more quickly than the site, for otherwise it’d be 5,000 pages long. It is intended to be a bridge between this site, for example, and a full on college textbook. It will hopefully be completely done in the next couple of weeks.
Unfortunately there’s not enough time in the day to write content fast enough for some of my extra keen readers (which again, is a great thing), but I hope that this book will provide what these keen readers are looking for. I’d love to hear what you think about all of this, and I hope you continue to read the site regardless of if you want/like/read/buy the book. Once this is done, I’ll be pumping out lessons at my former pace, I promise!
No math in this post, just an apology. I’ve been completely out of action with the site for the last couple of weeks, and I apologize for that. Please don’t take it as me being gone for good. The reason for my lack of activity here is that I’ve been working–for months–on a set of lecture notes for a high school course that will be starting in about a week. It covers everything the site does, plus a lot of what the site will cover in its first, say, 200 lessons (yes, I plan on writing that much, eventually). The notes are obviously more dense (the course is three weeks long), but still just as accessible (although requiring a bit more work).
When I’m done writing them I plan on posting them to the site, in case there are any extra keen readers who’d like to see what some of this stuff is all about. I’ll be done with those in a few days, and afterwards I’ll get right back to writing lessons. My promise to you is that I’ll have two more lessons up within the next 8 days.
So sorry again for my absence, but please don’t take it as the new status quo. I plan on sticking with this for a long, long time, and writing LOTS more lessons.
It is likely that until this moment in time you’ve been convinced that we have only five senses: seeing, hearing, tasting, touching, and smelling. This is, after all, what we’ve been taught since, well, I don’t know, forever I guess. But why is this so? What do we mean by “a sense”, and what do these classifications even do for us? I’ll describe some things I’ve thought about in this regard, and some ways that I see it relating to the study of mathematics. In doing so, I’ll propose that we in fact have (at least?) six senses. Don’t worry though—this sixth sense has nothing to do with seeing dead people (I’m hoping a reference to a 90’s movie isn’t too obscure here).
Let me take as a starting point the following seemingly simple question: What defines a sense? Despite its four word delivery, this is actually quite a subtle issue. To avoid a lot of the subtlety and leave that for the philosophers, I think we can define a sense to be some faculty that we use to perceive our surroundings. Surely, all of the information that we gather from the external world is gathered in the form of (at least) one of the five faculties of sight, sound, taste, touch, and smell. Or is it?
Since “perception” is a very abstract and complex issue to pin down precisely, let me use a different definition of “sense” that is easier to handle (i.e., more concrete), and that also captures what we want from our intuitive ideas about our senses. Namely, let me define a sense to be a faculty with which we can access beauty. The astute reader might now remark that our notions of “beauty” are ill-defined as well, which I would agree with. Thus, let me use this definition as motivation for an even more concrete definition: a sense is a faculty through which we can experience pleasure. Now, there are still some terms here that are hard to define, but let us take this as our definition and see where it leads us.
Clearly I have limited our definition of “a sense” quite a bit, from being something that is used to perceive the external world (quite a general statement) to something that is used solely to experience pleasure (something quite specific). Let me first explain the utility of this definition, and then perhaps its motivation will be more clear.
The power of this definition comes from the fact that it is relatively concrete and tangible, yet we can still recover the five senses that we know and love. In other words, we gain the ability to make forward progress without sacrificing any generality. Thus, even though I’ve made the definition more narrow, we actually don’t really lose much (see more about this sort of reasoning in the post “What is Abstract Thought?“). Let me now clarify how it is that we can recover the five usual senses from this more narrow definition of a sense as “a faculty through which one can derive pleasure”.
We can derive pleasure from seeing a beautiful sunset or painting, hearing a Beethoven symphony or the sound of the rain pattering the windows amidst distant rolling thunder, tasting a chocolate soufflé or an icy beer on a hot day, touching the soft fur of a puppy or the lips of a lover, and smelling freshly baked chocolate chip cookies or the top of a baby’s head (yeah, you know what I’m talking about). Thus, we have properly recovered the five senses that we already know and love. Moreover, anything that we can perceive from the external world could in theory be used to derive pleasure, and thus focusing on pleasure is a useful (and pleasant) tool, as well as a completely general one.
But are there other ways than these to experience pleasure? I believe that there is precisely one other way—a sixth sense—to derive pleasure, and that this sixth possibility is perhaps the most useful of them. I believe that logic is a sixth sense with which we can perceive pleasure, and that believe it or not we are all already familiar with this sixth sense. Before I mention exactly how we’re already familiar with this sixth sense, let me clarify that I am associating the pleasure derived from a sense with the emotional pleasure derived from the use of that sense. In other words, I am equating the pleasure of smelling freshly baked chocolate chip cookies with the memory of being a child and smelling the cookies that mom baked every Saturday (or whatever your memory is), so that the pleasure comes from the pleasurable emotions that were brought about by the sense in question. It is the use of the sense of smell which instigates that pleasurable experience, and I claim that the use of logic can instigate equally if not more profound experiences of pleasure.
I claim that we all already do this. I.e., we already use and derive pleasure from our sixth sense, we just don’t see it as such (yet?). For example, whenever we think of “the infinite” we are deriving some kind of pleasure by purely logical means. We ponder the numbers and realize that there can be no “greatest” number, for we could always add 1 (or 10, or 2,000) to that number. By realizing that there can be no greatest number, we come to the conclusion that there must be infinitely many numbers. We are then forced to stand in the presence and wonder of a very beautiful mental construction of the infinite, having used nothing but our sixth faculty of logic.
This, I argue, is precisely what mathematics is—it is the art form whose beauty is accessed through our sixth sense, and whose medium is logic.
Music is the art of constructing beauty through the medium of sound. A musician and/or composer trains to be able to create such structures, and the listener derives pleasure from those constructions via one of her six senses. A chef trains to improve his ability to create beauty via the medium (and sense) of taste, and a painter does so with sight. (As for touch and smell, I don’t know, think of a masseuse/masseur and Febreze, respectively).
In exactly the same way, mathematicians construct beautiful structures, the only difference is that we look upon these structures not with our eyes or ears, but rather with our sixth faculty of logic. Moreover, the sheer existence of these structures often evokes immense pangs of pleasure in those who look upon them. This pleasure is just of a different sort than that of the other five senses, but the same can be said about any of the others as well—there is no doubt that sight and sound are too wildly different senses.
So there you have it, my proposal for the existence of a sixth sense. Don’t be scared to develop it, tune it, sharpen it, and master it—it might just be one of the most important senses we have.
I’d be shocked if you’ve ever come to this site and not once wondered what’s going on with that picture across the top of each page. Who is this old scholarly dude and why is he looking at me? What are the balls all about? Well, this post is meant to give you my own reasoning for it, and interpretation of it. You’re certainly free to have your own interpretation, and if yours differs from mine I’d love to hear about it! Accordingly, this post will be purely philosophical, but that’s why it’s a post and not a lesson—I reserve the blog posts for shamelessly waxing philosophical.
The picture is a cut-off version (because WordPress.com can’t fit the whole thing) of M.C. Escher’s “Three Spheres II”. I won’t explain the picture, because there’s nothing I could say about it that you couldn’t deduce by looking at it (“there are three spheres, one is white,…”). First and foremost, if you’re not familiar with Escher’s work, Google image him right now because the guy was a total genius. Many of his prints are highly motivated by mathematical ideas, and some of these ideas are rather sophisticated (hyperbolic geometry, periodic tilings, impossible shapes, etc.).
This print in particular is, or at least I view it to be, a representation of Roger Penrose’s extremely gorgeous philosophy of mind, math, and physics. In his view of things, the world is divided up into three “smaller” worlds: the mathematical world, the physical world, and the conscious world.
The mathematical world can be viewed in a similar way as one views Plato’s world of forms—i.e., of supreme and eternal perfection, containing the truest essence of things. The only difference now is that this world of “forms” is a purely mathematical one; our forms are now nothing but mathematical structures.
The physical world is precisely what it sounds like: the world consisting of physical objects like your computer, this cup of coffee I’m drinking, stars, galaxies, black holes, atoms, and your body. This is the world that knows about the laws of physics, and has all of its constituents abide by them.
Lastly, the conscious world is that which consists of those things which are pure manifestations of consciousness. Seeing as consciousness is still not very well understood and extremely difficult to define, I’ll simply leave the definition to whatever the reader feels is that “obvious” sense of “being conscious” that we all (presumably) experience.
The key idea behind this philosophy is that these three worlds are not independent of each other. Instead, they influence and interact with each other in ways that are hard to deem as anything other than miraculous. In particular, the conscious world is able to interact with the mathematical world, since (assuming that humans are conscious) we use our consciousness to discover various mathematical structures and to understand their various relationships with each other. I.e., when we “do math”, we’re really using our consciousness to access this “other world” of eternal mathematical structures. Additionally, the mathematical world influences the physical world due to the fact that, as far as we can see, there is mathematical consistency to the laws that govern the physical world. I.e., the physical world is somehow nothing but a “manifestation” of certain structures that exist in the mathematical world. Lastly, the physical world affects the conscious world because (again, only as far we currently know), conscious things require a physical background to be implemented. I.e., whatever it is that makes us (or anything) conscious lies somewhere in our brains, and/or our bodies, and/or somewhere in the physical universe. In this way, these three worlds are completely interconnected in precisely the way that figure 1 depicts.
The reason I find this philosophy so pleasing is that each link interconnecting one world to the next is in no way obvious, or necessary. I cannot think of, nor have I encountered, any convincing argument for why any of these links had to be there. There is no reason that the laws of physics have to be mathematically consistent (assuming they are), or even that mathematics would be the proper language for describing them. One could plausibly imagine a world of genuine chaos (and indeed, some people do believe this to be our own physical world at the deepest levels, for which there is no evidence to either support or deny at this point). Moreover, it is in no way obvious that consciousness, whatever it is, has to use some physical object (like a brain) as a medium. We could perfectly well imagine some consciousness “floating around” out there which is not “tied down” to a physical embodiment. And lastly, there’s no a priori reason why anything that is conscious has to be able to know how to do math. I.e., it is perfectly reasonable to assume that something could be conscious without ever being able to develop some sophisticated logical machinery through which to access the world of math (and therefore the world of physics, by the transitive property of figure 1). The fact that all of these links appear to exist, at least to an extent, is incredible (in my view).
Note, however, that there are more to these worlds than their ability to influence “the next” world, in that there is more that our consciousness can do than just math, and there is more that math can do than just describe the laws of physics, and there is more that the physical world can do than just give rise to consciousness. This is embodied in figure 1 by the fact that only a part of each sphere is used to “hit” the next sphere.
Now let’s get back to the picture on each page of the site. What I see (and feel free to disagree) are these three worlds, sitting on M.C. Escher’s desk. The middle sphere, containing a reflection of Escher, is the conscious sphere (this is obvious, since it’s the one with a human “in” it!). The sphere on the left (as viewed from the artist) is the physical world, as it appears to be made of glass, a very “physical” object. And the sphere on the right is the mathematical world—completely clean and perfect. Moreover, we can see the reflection of both of these worlds in the middle sphere, which is the conscious world. In my mind, this signifies the fact that while the arrow goes from consciousness to mathematics, we can still posit the existence of the physical world via our conscious world. In other words, each world “knows about” the other two, but we can only “see” out of the conscious world, as that is in some clear way the world that we interact with the most.
I could write forever on this, but I think I’ll stop here. Now you at least have some explanation as to why this picture was chosen—whether you agree with this interpretation or not is entirely up to you. I have no pretentions of believing that this is anything but an untestable and unprovable philosophy. Regardless, this set of ideas has given me a very meaningful relationship with my craft (mathematical physics), and maybe others will find some beauty in this way of viewing things as well!
In this note we’ll introduce the notion of a “vacuous statement”—a statement that is true, but completely devoid of meaning. In particular, we’ll learn what it means for some statement to be “vacuously true”. Statements that are vacuously true come up from time to time in mathematics, and their existence (and truth) often are crucial for certain mathematical constructions to be logically sound. For example, we made use of a vacuous truth in our lesson on subsets, where we noticed that the empty set is a subset of every set. This is a great example of a vacuous statement, so let’s explore it in detail again.
Let us first remind ourselves briefly about subsets. Recall that if we have some set A, then a subset of A is some set whose every element is also in A. If the enemy hands me some set B and asks me if it’s a subset of A, all I need to do is consider every single element in B and ask if it’s also an element of A. If the answer is yes for every element in B, then the answer to the enemy’s question is yes. Suppose, however, that the enemy handed me the empty set and asked if it was a subset of A. What would be my answer? Well, what I would have to do is look at every element of the empty set and ask if it is also an element of A. But there are no elements in the empty set. Thus the statement “every element in the empty set is also in A” is true simply because there are no elements in the empty set to even consider!
Let’s take another example, which will hopefully make this notion clearer. This example won’t be as mathematically precise, but it will hopefully bring out the essential features of a vacuous statement, thus making the mathematically precise ones easier to handle. Consider the following statement: “Whenever there are cows on the moon, I can fly”. (I know this sounds totally absurd, but it’s actually mathematically relevant!) Now, I know that I can’t fly (despite my wishes). You know I can’t fly. Thus the statement “I can fly” is certainly false. However, it happens to be true that I can fly, provided that there are cows on the moon. Namely, every time in the history of the universe that there has been a cow on the moon, I have had the ability to fly. Of course (presumably), there has never been a cow on the moon. Also of course (at least I’m quite sure), I have never been able to fly. Thus it is indeed the case that the presence of cows on the moon has coincided perfectly with the instances of me being able to fly. Accordingly, the statement “Whenever there are cows on the moon, I can fly” is indeed true!
Now before you go crazy proving statements like “As long as pigs can speak French, Lebron is better than Kobe” (which would be true only in this circumstance), I should warn you that there is a reason why we call these statements “vacuous”. Namely, they are devoid of any content (like a vacuum). Thus, even though they’re true, they don’t tell us anything new. No one will be handed a Fields Medal for proving a vacuous statement, simply because a) they’re automatically true, and b) they don’t give us any new information. Just because I can prove that Lebron is better than Kobe whenever a pig can speak French, this does not mean that I can prove it in any scenario that might actually be relevant to the real world. The truth of the statement just kind of sits “out there” lacking any real substance. Nonetheless, the validity of some vacuous statements is important for certain mathematical results, and certain logical consistency. In particular, we used the fact that the empty set is a subset of every set (even itself) in order to establish the general pattern first described in lesson 3, and proved in lesson 15. These manipulations of the empty set will also come into play when we define topologies, as well as in category theory. Thus, despite the fact that the statement “the empty set is a subset of every set” is only vacuously true, its truth is necessary for much of the logical consistency of mathematics.
Now go have fun owning your friends in arguments by inserting various vacuously true statements in such a way that your argument is untouchable. Perfect this craft and you could seriously consider a career in politics!
One reason why people are scared of/hate mathematics is that once people see an expression like
they run and hide. Let us step back for a moment and try to see what’s really going on here, so that we can better understand this fear, and possibly do away with some of it.
The above expression is scary not because the ideas inherently hidden within it are bad or ugly, but rather only because we do not understand the notation of the expression. In other words, we don’t understand the chosen symbols used to represent the ideas. It might as well be Egyptian hieroglyphics for all we know. In fact, math could have been written down in Egyptian hieroglyphics and its logical content would still be exactly the same. Why is this? To answer this question, we need to better understand the role that notation plays for us in mathematics. In doing so, we’ll understand one of the many reasons why math is in fact not so scary.
I’ll begin with a question:
What do I mean when I write down the number 3?
Note that I’m not asking “what is the number 3”, but rather I am asking what the meaning of the symbol “3” is. This is a subtle but extremely important issue. The symbol 3 is shorthand notation for the abstract concept of “three” which we keep in our heads. Thus, when I write down 3 and you read 3, I can know that we’re keeping the same abstract idea in our minds. In other words, the symbol 3 means nothing without the abstract meaning that we give it. Moreover, it is simply a shorter, more concise way of keeping track of the abstract notion of “three-ness”. We could have written any other symbol in its place, and its abstract meaning would be the same.
Let us take an example that is even closer to home. Consider, for example, the words on this page. You’re looking at this strange concoction of squiggly and straight lines, interspersed with blank white space, and you’re deriving from it certain meaning. Simply put, you’re reading. You’re extracting meaning from the abstract ideas that are represented by a relatively arbitrary mixture of symbols. Moreover, the meaning derived from these symbols is independent of the symbols themselves, i.e., we could have all agreed to use other symbols and the abstract ideas would be left intact, so long as we agreed on what the symbols meant. Therein lies the creation of new/different languages!
In addition to representing words (abstract ideas) by symbols on a page, we can define new words based on old words. In doing so, we can have shorter ways of expressing more and more meaning. For example, we defined the word “bus” so that we could encapsulate all of the meaning of “bigger version of a car, designed for transporting large amounts of people” without having to always write the whole thing down. Thus, when you read the word “bus” you automatically associate it with this larger meaning, and the word “bus” is just a convenient abbreviation. We then can use this word to define a new word, namely “school bus”. This way we can simply say “school bus” instead of always having to say “bigger version of a car, designed for transporting large amounts of people to and from school”.
Math is no different. We use symbols to wrap up more and more meaning into less and less writing. This has no effect on the logical content of a given mathematical statement or equation—notation is simply used so that we humans can communicate mathematics to each other. Thus, if you can read (which I’m assuming you can since if you can’t you’ll never know that I’m assuming you can) then you are already doing exactly what you were scared to do by looking at the above equation—namely, deriving abstract information from a language designed to capture and emphasize certain ideas.
Here’s a quick way to become less scared of math. Consider the term “symplectic manifold”. There is a decent chance the reader has no idea what a symplectic manifold is, and therefore is confused, and therefore is worried that he’ll never understand math, and therefore is scared of math forever, and therefore tries to learn guitar instead, and therefore becomes that guy at parties always playing the guitar and pissing people off. I’m here to argue that there’s a better way.
As mentioned before, the words “symplectic manifold” are just abbreviations for other constructions, which are in turn abbreviations for other constructions—just like “school bus” (and just like the symbols in the expression at the top of the page—each standing for some larger, well-constructed idea). In fact, a symplectic manifold is nothing but a differentiable manifold with some additional properties, and a differentiable manifold is nothing but a manifold with some additional properties, and a manifold is nothing but a topological space with some additional properties, and a topological space is nothing but a set with some additional properties, and we study sets in the second lesson of this site, requiring no further background! Of course I’m not trying to say that a symplectic manifold is a particularly easy thing to study, but I am saying that there’s nothing to be inherently scared of. If you can make it through lessons 1 and 2 (and lesson 1 is hardly even a lesson), then you’re well on your way to learning what a symplectic manifold is. BOOM. All of mathematics is now at your fingertips…