Your Sixth Sense

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.

 

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The Three Worlds

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.

Figure 1  (Drawn by Roger Penrose himself)

Figure 1 (Drawn by Roger Penrose himself)

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!

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Meaningless Truth

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!

Posted in Mathematics | 6 Comments

Note on Notation

One reason why people are scared of/hate mathematics is that once people see an expression like

:A_0(x_1)\cdots A_0(x_n):\ = \sum_{X\subset\{1,...,n\}}\prod_{i\in X}A_+(x_i)\prod_{j\in CX}A_-(x_j)

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…

Posted in Mathematics | 19 Comments

Proof by Contradiction

In this post we’ll learn one of the most elegant and beautiful logical weapons that mathematicians can wield.  Every mathematician develops a toolbox for proving theorems, and this is one of the most powerful.  Its Latin name is reductio ad absurdum, and its English name is proof by contradiction.

How it works

Suppose we’re trying to prove that a statement (call it A) is true.  Also suppose that there are other statements that have already been proven true.  It doesn’t really matter what these other statements are, except for the fact that they have already been proven completely true.

What we do is suppose that the statement A is false.  There is nothing wrong with supposing something, so long as we never confuse it with something that is proven false.  Now, from this supposition we can start to derive other “valid” statements (scare quotes here to remind us that the subsequent statements are valid only if the supposition is correct (i.e., if A is actually false)).  Using logical deductions from our supposition, we want to arrive at a conclusion that contradicts something else that we know is true.

For example, call one of our already-proven-true statements B, and then suppose that A is false.  If we can use this supposition to show that B is false, then we have a contradiction because B is both true (already proven true) and false (just proven false)!  This is clearly not okay, and it tells us that the initial assumption was wrong.  What was the initial assumption?  That A was false.  Thus, A must be true!  Think about this one, it’s important.  In the meantime, here is a very trivial example (a better example is the proof of infinity primes).

Example

Suppose it is already proven that 0=0, and 0 doesn’t equal anything else.  I.e., zero only equals zero.  (This is obvious, but remember that we’re just looking at the logic here, not the math).  Suppose it is also known that we can add and subtract numbers in the normal way.  We can now ask the following trivial question: does 1=2?  Clearly the answer is no, but how can we prove it?  (obviousness is not a mathematical proof).

First, we assume the opposite.  Suppose that 1=2.  If that’s the case, then let’s see what happens when we subtract 1 from both sides (there’s nothing wrong with doing that).  Then 0=1.  But, we already know that zero equals zero and nothing else!  Therefore the conclusion that we’ve drawn here, namely that 0=1, is a contradiction!  This means our supposition was impossible.  What was our supposition?  That 1=2.  Thus it is the case that 1 does not equal 2.

Yes, this is trivial, but we use this method all the time for proving cooler stuff.  For example, it’s used in lesson 13 for showing that there are, in fact, at least two fundamentally different kinds of infinities, and again in lesson 14 to show that there are infinitely many different kinds of infinity!

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Infinity Primes

Prerequisites: Prime numbers, proof by contradiction

Here’s an awesome, easy, super clever, and really beautiful theorem/proof combination (which I cannot take credit for (Euclid can, though)).  Once you’ve learned what a prime number is and how to use proof by contradiction, you’ll be good to go (seriously, that’s it).

The beginning of the list of primes goes as follows: \{2, 3, 5, 7, 11, 13, 17, 19, 23, ...\}.  You can write out a thousand more primes if you’d like, but I’m sure you get the picture.  The first natural question to ask is the following: does this list go on forever?  In other words, are there infinitely many primes?  It is clear that an equivalent question would be to ask if there is a largest prime (clearly there is a smallest prime (which is 2), so there is a finite list of primes if and only if there is a largest prime).

Now, there is no reason that there has to be an infinite amount of primes.  As numbers get bigger they become more and more divisible (i.e., there are more numbers that are smaller than them, and therefore more candidates for things that could divide them), so it might be plausible for there to be a cut-off point above which every number can be divided and is therefore not prime.

How might we go about proving a statement regarding infinity?  If the enemy listed a whole bunch of consecutive non-prime numbers, it might provide evidence for the fact that there is a “largest prime”, but that is in no way a proof.  For I could just take the largest number that the enemy gave me and multiply it by itself a hundred billion billion trillion times and tell him to prove that we didn’t “miss” any primes.  What do we do?

Proof by contradiction

We will suppose that there is a largest prime, and arrive at a contradiction.  This contradiction will force us to draw the opposite conclusion, thus proving our claim.  Here we go.

Theorem: There are infinitely many prime numbers.

Proof: Suppose there aren’t.  Then there is a “largest prime”.  We don’t know what this prime is, but we know it’s out there (because we just supposed it exists), so let’s call it P.  Then the entire list of primelooks something like \{2, 3, 5, 7, 11, ..., P\}.  This hypothetical list hypothetically contains all the primes.  If the enemy gives us this list, we can define a new number in the following way: Let N = (2\times 3\times 5\times 7\times 11...\times P)+1.  In other words, N is the number defined by multiplying all of the prime numbers together and then adding 1.  This number would be huge, but it is finite (because our assumption says that there are only a finite amount of primes).

Now we can ask whether or not N is prime.  Remember, it is one or the other, but not both and not neither.

Well, clearly it can’t be prime, because it’s larger than P and we supposed that P was the largest prime.  Thus if N were larger than P and prime, P wouldn’t be the largest prime, which is a contradiction!  So then P must be not prime.  Now if N is not prime, then N must be divisible by a prime number.  This is because N is divisible by something (because it’s not prime), and this something is then either prime or not prime.  If this something is prime, then we’ve shown that N is divisible by a prime.  If this something is not prime, then this something is in turn divisible by something else.  We now just apply the same logic to this “something else”, and eventually we’ll hit a prime.  Thus, since we know N is not prime, it must be divisible by a prime.

Recall though that all of the primes are in our list \{2, 3, 5, 7, ..., P\}.  If we divide N by any of these numbers, we’ll always get a remainder of 1 because N was defined to be the product of multiplying all of these numbers together, and then adding 1.  Thus none of these numbers divides N evenly, so N is not divisible by a prime.  Since we know that every non-prime number is divisible by a prime, N must not be non-prime (sorry for the double negative).  But this means that N is both not prime, and not not prime (double negative)!  This is impossible, since we already know that every number is either prime or not prime.  Thus we’ve reached a complete contradiction, which means that our initial supposition was wrong.  And what was our initial supposition?  It was that there was some largest prime P.  Thus, there can’t be such a “largest prime”, and the only way that this can be true is if the primes go on forever.  Thus there are infinitely many primes!

QED (which means “you just finished the proof and you’re a boss!” in Latin).

Posted in Mathematics | 3 Comments

What is abstract thought?

In various posts throughout this site we have discussed the notion of abstract thought.  Abstract thought is the primary tool that mathematicians use in practicing their art form.  In fact, one could reasonably say that mathematics is the art of pure abstract thought.  The only problem with this, however, is the potential circularity in the reasoning: math is abstract thought, and abstract thought is that which mathematicians do.  Thus, we need to try to attack abstract thought directly.  This is no easy task.

Generalizing: Driving your car = skydiving

Abstract thought is very closely related to the mental process of generalizing.  Another way to think of this is that abstract thought is that which explores what something really “is”.

For example, I am currently drinking a glass of water.  I can generalize this in an infinite number of different ways, but here are a few: I am drinking water, I am consuming water in some way, I am nourishing my body, and I am doing something.  In each of these cases, the statement “I am drinking water” is only a special case.  Namely, if I am drinking water, then I am certainly drinking, I am nourishing my body, I am consuming water in some way, and I am doing something.  The converse is not necessarily true.  For example, I could be drinking orange juice, in which case “I am drinking” is true, but “I am drinking water” is not.  Similar counter examples can be found for the other generalizations.

This generalization is nice because anything that I can say about drinking, or nourishing my body, or consuming water in some way, or doing something, will also be true in the case of drinking water.  For example, if I say “drinking is good,” then it will also be true that “drinking water is good”, because drinking water is a special case of drinking.  You can think up several different examples, and it’s usually pretty fun to do so.

More than just generalizing

Abstract thought also includes the act of appraising the value of a certain generalization.  In other words, it is possible to “over-generalize” and reach a point of generalization that is no longer fruitful.  In the above examples, “I am doing something” would be a point of over-generalization in my opinion.  This is because if I want to make a meaningful generalization of “I am drinking water,” then I don’t want to generalize to the point that “drinking water” and “fighting a gorilla” are both special cases of the same thing.

This is, of course, a matter of taste in this instance.  In mathematics, however, the extent to which an idea is generalized is immensely important for making meaningful progress.  For example, if I took an object that could be generalized to a group (which is a very special type of set, with some added structure) and “over-generalized” it to a generic set (because a group is a set, but a lot of sets are not groups), then I will have lost a lot of meaningful information about the object.  Yes, it is true that anything that I prove to be true about a set is true about a group, but there are likely many important things that I can prove about a group that I can’t prove about a set, and I therefore might not want to generalize everything to a set.

This is how abstract thought is more than mere generalization.  It is also the intangible knowledge of when to stop generalizing.  We will often see the power of this type of thought, and indeed it is abstract thought that makes all of math “go”.  For now, however, it might be fun to try to generalize everything in your life to an almost comical degree.  For example, driving your car is a special case of driving a vehicle, which is a special case of driving, which is a special case of transporting yourself.   Sky-diving is also a special case of transporting yourself (transporting yourself from a plane to the ground, quickly).  Thus, driving and skydiving are, in precisely this way, the same!

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What is a mathematical structure?

In the post “What is math?”, we described mathematics as the art of creating and exploring mathematical structures.  It is not unlikely, however, that the reader is slightly unfamiliar with the notion of a mathematical structure.  If this is the case, then our definition of mathematics is rather unsatisfying.  This post aims to rectify this.

Structures in general

When we think of a structure in the everyday sense, we might think of buildings, houses, and bridges.  We may also think of a structure as a more abstract object involving some form of complex organization.  The plot of a movie, a musical composition, and government bureaucracies all are structures in some sense.  All of these are instances in which small sub-structures are organized in ways to create larger, more complicated patterns.  A building is nothing but the complicated organization of smaller sub-structures such as bricks, cement, wood, and iron.  A musical composition is a complicated organization of melodies and harmonies, which are in turn complicated organizations of notes and rhythms.

Math Structures

Math is no different.  A mathematical structure is nothing but a (more or less) complicated organization of smaller, more fundamental mathematical substructures.  Numbers are one kind of structure, and they can be used to build bigger structures like vectors and matrices (the definitions for which will be posted in the future).

There are plenty of other kinds of mathematical structures that exist in a rather fundamental way, and that can be used to build other remarkably beautiful structures.  Sets and functions are both incredibly fundamental in mathematics, and they can be used to build crazy things like topological spaces (again, which haven’t been defined (yet) on this site, but will be soon).  Sets and functions can also be tools for exploring different types of infinities.

Building your castle

One of several mathematical castles that you can build for yourself!

Studying math is like building a castle in your head.  When building a castle, you first must learn to build a brick, and once that is mastered, you can use it to build a wall.  Once you can build a wall, you can build a tower.  Stronger bricks allow for higher walls and bigger towers.  Additionally, powerful tools allow you to build faster and more efficiently.

The beauty of a mathematical structure comes from its ability to have larger structures built from it.  Certain mathematical concepts allow for faster building than others.  For example, a mathematician will find a mathematical crane much more useful than a mathematical wheelbarrow.

While you were in high school, you likely only learned about one type of structure—those that could be built up from numbers.  Although some of this is interesting, the real beauty of math lies in the flexibility and deep interconnectedness of various kinds of mathematical structure.  We explore several of these structures throughout this site, and I believe that once you start on your castle, you’ll never want to stop.

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What is mathematics?

Math isn’t what you thought it was.

You might think that math goes something like this:

Calculate or simplify the following expressions:

  1.  535\times 45
  2. 12^3
  3. the derivative of f(x)=3x^4 + 25x^3 + 5\mathrm{sin}(x)
  4. (x+5)^3

Well, that all sucks.  No one wants to do any of that.  From this, one might think that math is just remembering the right formula, or the right steps, and then having the patience and focus to follow these steps for several minutes without making a mistake.  It involves tediously writing numbers and letters all over the paper in exactly the way that your teacher told you would work.  We often have no real idea as to why we’re doing this, or any knowledge of the vast logical constructions that make it all possible—we’re just calculating.  This post will explain that there is another kind of math—one that focuses primarily on the abstract foundations and the deep logical reasoning that goes in to establishing the math that we’re used to doing every day.  To begin exploring this “other” type of math first-hand, head on over to the lessons!

Two types of math.

I am going to refer to the above kind of math as “calculation”.  I.e., when we’re answering questions 1-4 above, we’re not doing our “mathematics” homework, but rather our “calculation” homework.  I will reserve the term “real mathematics” for what I mean by this “other world” of math.

What is “real” mathematics?   There are volumes written about this question, and I will only spend a few sentences on it here.  As you spend time reading through the lessons and exposing yourself to what I call “real” math, you’ll form your own idea of what mathematics is.  We can, however, make a little progress right here and now.

“Real” mathematics (or just mathematics) is the art of creating, understanding, and exploring the relationships between various mathematical* structures.  Mathematics is the art of making mathematics, by exploring and  understanding mathematical structures.  (Numbers are one such structure, but there are infinitely many more.  We will explore mathematical structures more in depth later, and many examples of such structures can be found on this site.)

There is indeed a good amount of what I’m calling “real” mathematics hidden inside of a standard high school math curriculum, and those who like math tend to be drawn to these bits.  But these parts of the curriculum are often drowned out by the emphasis on memorization, rote learning, and repetition.  Questions 1-4 above lie on top of a very elegant logical framework, but we often completely ignore this in order to get on with our calculations.

What do mathematicians do?

A mathematician does not spend his day calculating 2+2 or “the derivative of f(x)=3x^4 + 25x^3 + 5\mathrm{sin}(x)”.  He or she instead thinks about why and precisely when it is the case that 2+2=4, or what a derivative really means and what it can be used for.  The more math you learn, the more you realize that 2+2 does not always equal 4 (this will be explained when we describe “clock arithmetic”).  The remarkable logical foundation on which the true statement “2+2=4” stands was built by mathematicians who were doing “real” mathematics.  The actual process of calculating the result of 2+2 (and other similar calculations) is important for students to learn, but it is not the kind of thinking that goes into much of mathematics.

Everyone, from high school students to professional mathematicians, hates calculating derivatives.  The difference is that professional mathematicians have been exposed to the logical and rigorous foundations that make these calculations possible.  They’ve studied what derivatives are, where they come from, and what they lead to, and these kinds of considerations are what keeps them in the business of studying math.  Additionally, they find a deeper and more subtle beauty in the calculations themselves.  This site aims to introduce its readers to this type of math—what I’m calling “real” (as opposed to calculational)—and to explore some of the gorgeous subtleties that come along with it.

*You might be worried that I used the term “mathematical” in the definition of mathematics, but I have reason to do this.  A mathematical structure is a thing, and I am using it to describe what mathematics is as an art form.  In other words, there would be no problem saying that “painting is the art of making paintings”, and that is essentially what I’ve done here.

So true.
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