Let’s begin with a short practical experiment. Pick up a pen, or whatever similar sized object is handy, hold it a short distance above the ground, and drop it. The result — that the pen falls to the ground — is not a surprising one. The point of the experiment was not to note the result, however, but rather to note our lack of surprise at it. We expect the pen to fall to the ground; our expectation is based not on knowledge of the future however, but on abstraction from past experience. Chambers Dictionary defines “abstract”, the verb, to mean “to generalize about something from particular instances”, and it is precisely via this action that we come to expect the pen to fall to the ground. By synthesis of many previous instances of objects falling when we drop them, we have generalized the rule that things will always fall when we drop them*. We make this abstraction so instinctively, and take it so completely for granted, that it is worth dwelling on it for a moment so we can see how remarkable it actually is.

The circumstances surrounding each and every instance of you observing an object falling to the ground are quite unique. Were it not for our brains’ natural tendency to try to link together our experiences into some kind of narrative we would be left contemplating each dropped object as an entirely distinct instance, and be in no position to have any expectation as to what will happen; each time it would be an entirely new case. In our minds we have, rather than a vast array of disjoint and distinct instances, a single principle that knits together the common elements and generalises to new circumstances that share the same common elements of those past experiences. This is something we do unconsciously and automatically: our brains hunt for patterns in the world, generalises those patterns, and expects them to continue. Indeed, almost all our expectations are a result of such inductive knowledge and abstraction. Abstraction is fundamental to our experience.

Of course not all abstractions are correct. Our minds are constantly on the hunt for possible patterns, and don’t always pick out valid ones. The classic example is the Christmas goose, who, every day of the year has found the arrival of the farmer results in the goose getting fed — until Christmas day when the abstracted rule that “Farmer implies food” meets a painful end. Even if we find abstractions that are ostensibly correct, that doesn’t mean they’re ideal. A perfectly valid abstraction of dropping objects, for instance, would be the rule that a dropped object always accelerates away from your hand. Certainly that is true, but it leaves out what might be considered an important common property of things being dropped: the direction in which the dropped object accelerates. Alternatively we could note that different objects, say a feather compared to a stone, behave very differently when dropped and arrive at a vast array of rules, one for each different kind of object.

This leads to the dilemma of finding the most effective or efficient abstraction — the abstraction that most consistently produces the results you want with the least effort. That is where science comes in: it is a systematic effort to refine our abstracted rules and principles, and continually check them for consistency. We may, if we like, think of the different sciences as arising from the “results we want” clause of our definition of effective abstraction. A biologist tends to work with different abstractions (generally on very different scales) than a physicist because the sorts of results they are interested in determining are rather different.

This is all very interesting, but you are probably starting to wonder what any of it has to do with mathematics. The answer is that mathematics relies on precisely this sort of abstraction that is so integral to our experience of the world. Mathematics simply attempts to take the abstraction as far as it possibly can. Part of what makes mathematics “difficult” is that it tends to pile abstraction upon abstraction. That is to say, after developing a particular abstraction it is common for mathematics to then study that abstraction and, upon finding common properties when dealing with that abstraction, generalise that commonality into a new abstraction; mathematics develops abstractions not only from a synthesis of experiences of the external world, but also via synthesis of properties of existing abstractions. This layering means that unless you’ve gotten a good grasp of the preceding level of abstraction, the current one can be extremely hard to follow. Once you’ve wandered off the path, so to speak, it can be difficult to find your way back.

The other problem that people tend to face, when learning mathematics, is that as you pile up abstractions and climb higher, and hence more distant from everyday experience, intuition becomes less and less helpful, and an increasing degree of pedantry is required. To give an example of what I mean by this, lets take a detailed look at a mathematical abstraction that almost everyone takes for granted: numbers.

Natural numbers (also known as counting numbers) are one of those remarkable abstractions, like objects falling to the ground, that we take for granted. Natural numbers are, however, a very abstract concept – they simply don’t exist in the external world outside your own head. If you think otherwise, I challenge you to show me where the number 3 exists. You can point to some collection of 3 things, but that is only ever a *particular instance* that possesses the property common to all the particular instances from which the number 3 is generalised. Because we take natural numbers for granted we tend to make assumptions about how they work without thinking through the details. For instance we all know that 1 + 1 = 2 – but does it? Consider a raindrop running down a windowpane. Another raindrop can run down to meet it and the separate raindrops will merge into a single raindrop. One raindrop, plus another raindrop, results in one raindrop: 1 + 1 = 1. The correct response to this challenge to common sense is to say “but that’s not what I mean by 1 + 1 = 2” and going on to explain why this particular example doesn’t qualify. This, however, raises the question of what exactly we do mean when we say that 1 + 1 = 2. To properly specify what we mean, and rule out examples like putting 1 rabbit plus another rabbit in a box and (eventually) ending up with more than 2 rabbits, is rather harder than you might think, and requires a lot of pedantry. I’ll quote the always lucid Bertrand Russell** to explain exactly what we mean when we say that 1 + 1 = 2:

Omitting some niceties, the proposition ‘1 + 1 = 2’ can be interpreted as follows.

We shall say that φ is a unit property if it has the two following properties:

- there is an object
ahaving the property φ;- whatever property
fmay be, and whatever objectxmay be, ifahas propertyfandxdoes not, thenxdoes not have the property φ.We shall say that χ is a dual property if there is an object

csuch that there is an objectdsuch that:

- there is a property
Fbelonging tocbut not tod;chas the property χ anddhas the property χ;- whatever properties
fandgmay be, and whatever objectxmay be, ifchas the propertyfanddhas the propertygandxhas neither, thenxdoes not have property χ.We can now enunciate ‘1 + 1 = 2’ as follows: If φ and ψ are unit properties, and there is an object which has property φ but not the property ψ, then “φ or ψ” is a dual property.

It is a tribute to the giant intellects of school children that they grasp this great truth so readily.

We define ‘1’ as being the property of being a unit property and ‘2’ as being the property of being a dual property.

The point of this rigmarole is to show that ‘1 + 1 = 2’ can be enunciated without mention of either ‘1’ or ‘2’. The point may become clearer if we take an illustration. Suppose Mr A has one son and one daughter. It is required to prove that he has two children. We intend to state the premise and the conclusion in a way not involving the words ‘one’ or ‘two’.

We translate the above general statement by putting:

φ

x. = .xis a son of Mr A,ψ

x. = .xis a daughter of Mr A.Then there is an object having the property φ, namely Mr A junior; whatever

xmay be, if it has some property that Mr A junior does not have, it is not Mr A junior, and therefore not a son of Mr A senior. This is what we mean by saying that ‘being a son of Mr A’ is a unit property. Similarly ‘being a daughter of Mr A’ is a unit property. Now consider the property ‘being a son or daughter of Mr A’, which we will call χ. There are objects, the son and daughter, of which (1) the son has the property of being male, which the daughter has not; (2) the son has the property χ and the daughter has the property χ; (3) ifxis an object which lacks some property possessed by the son and also some property possessed by the daughter, thenxis not a son or a daughter of Mr A. It follows that χ is a dual property. In short … a man who has one son and one daughter has two children.

As you can see, once we try to be specific about exactly what we mean, even simple facts that we assume to be self-evident become mired in technical detail. For the most part people have a sufficiently solid intuitive grasp of concepts like “number” and “addition” that they can see that 1 + 1 = 2 without having to worry about the technical pedantry. The more abstractions we pile atop one another, however, the less intuition people have about the concepts involved and it becomes increasingly important to spell things out explicitly. In fact much of modern mathematics has reached the point where it is sufficiently far divorced from everyday experience that common intuition is counterproductive, and leads to false conclusions. If you think that sounds silly then consider that modern physics has also passed this threshold — few people can claim quantum mechanics to be intuitive. Our experience of the world is actually remarkably narrow, and thus so is our intuition.

At about this point I imagine some readers are wondering, if it is so easy to get lost and mired in technicality, why bother with all of these abstractions? I like to think of this process of increasing abstraction as akin to a road deep into the mountains. At times the road can look awfully steep. At other times, to avoid a sheer climb, the road is forced to take a tortuous winding path. As you progress deeper into the mountains, however, you will find places where the road opens out to present you with a glorious vista looking out over where you have come from. Each new view allows an ever broader view of the landscape, allowing you to see further and more clearly, while also seeing all the other different roads that all lead to this same peak. It is these unexpected moments, upon rounding a corner, of beauty, and clarity and insight, that, to me, make the study of mathematics worthwhile. I hope that, in this journey into the interior of mathematics, I can impart to you some of those moments of beauty and wonder.

* Note that in practice some expectations have been, to some extent, pre-wired into our brains, and this particular discussion is simply an illustrative example of the general concept of abstraction.

** From *Essays in Analysis*, Section V, Chapter 15: Is Mathematics Purely Linguistic, pages 301 and 302.

December 12, 2006 at 10:14 pm |

[…] My task, then, is to provide you with the necessary experiences in the mathematical landscape; to provide context for the insights that are to follow. In the previous entry, On Abstraction, I discussed the process of abstraction, and how mathematics builds up layer upon layer of abstraction. The road we must take, the slow road, is the path that winds its way through these layers. Each layer is, in a sense, a small plateau amidst the mountains; to be explored before the next rise begins. […]

January 28, 2007 at 3:49 am |

[…] The Narrow Road » Blog Archive » On Abstraction Let’s begin with a short practical experiment. Pick up a pen, or whatever similar sized object is handy, hold it a short distance above the ground, and drop it. The result – that the pen falls to the ground – is not a surprising one. The point of the experiment was not to note the result, however, but rather to note our lack of surprise at it. Share and Enjoy:These icons link to social bookmarking sites where readers can share and discover new web pages. […]

March 5, 2007 at 12:12 am |

[…] As a mathematician there is a story I hear a lot. It tends to come up whenever I tell someone what I do for the first time, and they admit that they don’t really like, or aren’t very good at, mathematics. In almost every case, if I bother to ask (and these days I usually do), I find that the person, once upon a time, was good at and liked mathematics, but somewhere along the way they had a bad teacher, or struck a subject they couldn’t grasp at first, and fell a bit behind. From that point on their experiences of mathematics is a tale of woe: because mathematics piles layer upon layer, if you fall behind then you find yourself in a never ending game of catch-up, chasing a horizon that you never seem to reach; that can be very dispiriting and depressing. In the previous entries we have dealt with subjects (abstraction in general, and the abstraction of numbers) that most people have a natural intuitive grasp of, even if the details, once exposed, prove to be more complex than most people give them credit for. It is time to start looking at subjects that often prove to be early stumbling blocks for some people: fractions and algebra.There is a reason that these subjects give people pause when they first encounter them, and that is, quite simply, that they are difficult. They are difficult in that they represent another order of abstraction. Both fractions and elementary algebra must be built from, or abstracted from, the basic concept of numbers. Because of the sheer prevalence of numbers and counting in our lives from practically the moment we are born, people quickly develop a feel for this first, albeit dramatic, abstraction. It is when people encounter the next step, the next layer of abstraction, in the form of fractions and/or algebra, that they have to actively stretch their minds to embrace a significant abstraction for the first time. Most of us, having won this battle long ago, struggle to see the problem in hindsight — we might recall that we had trouble with the subject when we were younger, but would have a hard time saying why. We have developed the same sort of intuitive feel for fractions and algebra as we have for numbers and have forgotten that this is hard won knowledge. […]

March 27, 2007 at 8:55 pm |

[…] Let us tackle the Dichotomy first. To ease the arithmetic, let us assume that the moving body in question is traversing an interval of unit length (which we can always do, since we are at liberty to choose what distance we consider to be our base unit), and that it is travelling at a constant speed. We can show that, contrary to Zeno’s claim, the object can traverse this distance in some unit length of time (again, a matter of simply choosing an appropriate base unit) despite having to traverse an infinite number of shorter distances along the way. To see this, consider that, since the body is travelling at a constant speed, it would have to cover a distance of 1/2 in a time of 1/2, and before that it would cover a distance of 1/4 in a time of only 1/4, and so on. The key to resolving this is that the infinite sum 1/2 + 1/4 + 1/8 + 1/16 + … is equal to 1, and thus the infinite tasks can, indeed, be completed in finite time. This tends to be the point where most explanations stop, possibly with a little hand-waving and vague geometric argument about progressively cutting up a unit length. It is at this point, however, that our discussion really begins. You can make intuitive arguments as to why the sum turns out to be 1, but, given that we weren’t even that clear about what 1 + 1 = 2 means, a little more caution may be in order — particularly given that infinity is something completely outside our practical experience, so our intuitions about it are hardly trustworthy. […]

March 31, 2007 at 11:42 pm |

very good!

April 4, 2007 at 8:48 am |

Excellent topic.I am having trouble though at understanding the following :

We can now enunciate ‘1 + 1 = 2′ as follows: If φ and ψ are unit properties, and there is an object which has property φ but not the property ψ, then “φ or ψ” is a dual property.

as I would have added that “and there is another object that has property ψ but not the property φ” ; or having considered the fact that φ and ψ are unit properties means that there exist 2 objects of properties φ and ψ and by saying that “there is an object which has property φ but not the property ψ” we ensure they are different objects?

Thanks you very much for the great topics.

April 4, 2007 at 8:57 pm |

Alex: Your last statement is the right interpretation. Being a unit property ensures that an object having the property exists, thus φ and ψ being unit properties mean that there is an object with property φ and there is an object with property ψ. For a dual property we just need to ensure that we don’t have just one object which has both properties φ and ψ, and if the object that has property φ doesn’t have property ψ then by the second condition of unit properties any object that has property ψ can’t have property φ (though it’s little tricky to get that). If we drop into mostly logic notation it is easier to see:

The second condition of a unit property φ says that:

∀ f,x: f(a)∧¬f(x)→¬φ(x)

So in particular if x is an object such that ψ(x) is true (and thus ¬¬ψ(x) is true, assuming classical logic), and we let f be the property ¬ψ, we can get

¬ψ(a)∧¬¬ψ(x)→¬φ(x)

since we’ve been assured that ¬ψ(a) is true.

April 21, 2007 at 9:19 am |

Hello, you have a nice site, good LUCK!

April 21, 2007 at 4:41 pm |

this guy is good…lol amazing…

April 24, 2007 at 7:22 pm |

please keep this up! this is the way math should’ve been taught to me in school!!! i totally get this..

May 2, 2007 at 10:14 pm |

This is beautifully thought out and eloquently written. Thank you so much for an intelligent and thought-provoking insight into a staggeringly sumptuous world of wonder and delight.

I wish you the best of luck in this adventure, and sincerely hope that you can impart theories of calculus in an accordingly dazzling manner; it deserves an exquisite treatment.

May 13, 2007 at 9:24 am |

Hello, nice site. Good luck

May 18, 2007 at 9:35 am |

hello. very nice. good luck

May 29, 2007 at 11:47 pm |

I am delighted to see you undertaking this project. I’ve wished for such an introduction to Mathematics most of my life. As a computer scientist and someone generally interested in science and mathematics I am always trying to expand my understanding. Most mathematics texts read to me like the work of old-fashioned programmers who delight in low-level coding. I prefer a modular and incremental presentation of complex ideas.

I find your words and imagery both clear and beautiful. I find Russell’s convoluted and deadening. I would encourage you to replace the Russell quote with something of your own.

What I get from Russell’s torturous expression is that he is trying to say that counting presupposes distinctness, and then he generates a basis for that with a deliberately non-intuitive approach. Trying to kill intuition may have been desirable for his Principia Mathematica program, but it is very off-putting to a beginner! I would prefer something more like this:

(1) Let “x differs from y” mean some property p exists which holds for x and does not hold for y.

(2) Let “property p is a unit property” mean that

(2a) an entity a exists satisfying property p, and

(2b) for all entities x where “x differs from a”, x does not satisfy property p.

and so on. However, even after disentangling Russell’s expression, I’m not at all convinced that this is a good basis for understanding counting. I would prefer an approach involving one-to-one correspondence, as that is closer to how humans (and some other animals!) learn counting. I think also we draw on our geometrical notions of magnitude. Counting sits on a pleasant peak which can be reached by several easy (and some quite challenging) paths. The views from that peak are illuminating and many paths from there lead onwards and upwards!

July 4, 2007 at 5:24 am |

[…] Going all the way back to the first entry, On Abstraction, things start to get a little clearer however. As long as we view mathematics as a matter of making effective and powerful abstractions from the real world, rather than describing some platonic universe, having a choice of abstraction doesn’t seem so bad. We can choose how to interpret the continuum to suit our needs — indeed, we can even reject transfinite arithmetic and opt for the intuitionist conception of the continuum if we wish; we choose the abstraction that best suits our purposes for the moment. You could view it as little different than choosing to work at the genetic level as a molecular biologist instead of the considering subatomic particles as a physicist would: the level and manner of abstraction matters only with regard to the level and manner of detail you wish to obtain in the way of results. The more layers of abstraction we apply, the greater the chances of running into quandaries and choices; by abstracting away more and more detail, and by piling abstractions upon abstractions, we push further and further into the realm of pure possibility. This has the potential to lead us to strange and confusing trails, but it also gives us the power to see beyond our own limited horizons. In broadening our minds to embrace worlds of possibility we conceive of realities that transcend our conceptions, and probe our own reality in ways far beyond the limits evolution has shackled our perceptions with. […]

September 26, 2007 at 5:23 pm |

I do like that raindrop analogy.

May 16, 2008 at 3:30 am |

Great explanation of abstraction. Thank you.

March 12, 2010 at 7:37 pm |

[…] away from real world objects, and manipulate these abstractions to draw deep results, is vital. Abstraction is fundamental to mathematics; it is what gives mathematics both its power and its scope; it is the mechanism by which higher […]