Category Archives: math

math, understanding math

Stances

4 Comments

Philosophy

With the help of some colleagues, I am beginning to understand why I am bothered by most discussions of the philosophy of math.  Philosophers have a stance. Examples:

  • "Math objects are real but not physical."
  • "Mathematics consists of statements" (deducible from axioms, for example).
  • "Mathematics consists of physical activity in the brain."

And so on.  They defend their stances, and as a result of arguments occasionally refine them.  Or even change them radically.  The second part of this post talks about these three stances in a little more detail.

I have a different stance:  I want to gain a scientific understanding of the craft of doing math.

Given this stance, I don't understand how the example statements above help a scientific understanding.    Why would making a proclamation (taking a stance) whose meaning needs to be endlessly dissected help you know what math really is?

In fact if you think about (and argue with others about) any of the three, you can (and people have) come up with lots of subtle observations.  Now, some of those observations may in fact give you a starting point towards a scientific investigation, so taking stances may have some useful results.  But why not start with the specific observations?

Observe yourself and others doing math, noticing

  • specific behaviors that give you forward progress,
  • specific confusions that inhibit progress,
  • unwritten rules (good and bad) that you follow without noticing them,
  • intricate interactions beneath the surface of discourse about math,

and so on.  This may enable you to come up with scientifically testable claims about what happens when doing math.  A lot of work of this sort has already been done, and it is difficult work since much of doing math goes on in our brains and in our interactions with other mathematicians (among other things) without anyone being aware of it.   But it is well worth doing.

But you may object:  "I don't want to take your stance! I want to know what math really is."  Well, can we reliably find out anything about math in any way other than through scientific investigation?   [The preceding statement is not a stance, it is a rhetorical question.]

Analysis of three straw men

The three stances at the beginning of the post are not the only possible ones, so you may object that I have come up with some straw men that are easy to ridicule.  OK, come up with another stance and I will analyze it as well!

"I think math objects are real but not physical."  There are lots of ways of defining "real", but you have to define it in order to investigate the question scientifically.  My favorite is "they have consistent and repeated behavior" like physical objects, and this behavior causes specific modules in the brain that deal with physical objects to deal with math objects in an efficient way.  If you write two or three paragraphs about consistent and repeated behavior that make testable claims then you have a start towards scientifically understanding something about math.   But why talk about "real"?  Isn't "consistent and repeated behavior" more explicit?  (Making it more explicit it makes it easier to find fault with it and modify it or throw it out.  That's science.)

"Mathematics consists of statements".  Same kind of remark:  Define "statement".  (A recursively defined string of symbols?  An assertion with specific properties?)  Philosophers have thought about this a bunch.  So have logicians and computer scientists.  The concept of statement has really deep issues.  You can't approach the question of whether math "is" a bunch of statements until you get into those issues.  Of course, when you do you may come up with specific testable claims that are worth looking into.   But it seems to me that this sort of thinking has mostly resulted in people thinking philosophy of math is merely a matter of logic and set theory.  That point of view has been ruinous to the practice of math.

"Mathematics consists of physical patterns in the brain."   Well, physical events in the brain are certainly associated with doing math, and they are worth finding out about.  (Some progress has already been made.)  But what good is the proclamation: "Math consists of activity in the brain".   What does that mean?  Math "is" math texts and mathematical conversations as well as activity in the brain.   If you want to claim that the brain activity is somehow primary, that may be defendable, but you have to say how it is primary and what its relations are with written and oral discourse.  If you succeed in doing that, the statement "Math consists of activity in the brain" becomes superfluous.

Send to Kindle
math

Curvature

4 Comments

This post is the result of my first experiment with the capability of including TeX in WordPress blogs (that capability is the reason I switched from Blogger).  This article will eventually appear as an example in abstractmath.org with lots of links to posts in the website that are germane to reading and understanding the article.

parabolasmall

Measuring bending

The curve y=x^2 (graph above) has a fairly sharp bend near the origin but as you move away from the origin in either direction it looks more and more like a straight line.   To measure this bendiness, each curvey=f(x) in the real plane has an associated curvature function  that measures how bent the curve is at each point.  (For this to work, f must have first and second derivatives.)  By definition, the curvature ofy=f(x) at x is given by:

\kappa_f(x)=\frac{\left|f''(x)\right|}{\left(f'(x)^2+1\right)^{3/2}}

The curvature function has the following three properties:

  • The curvature at any point on a straight line is 0.
  • The curvature at any point on a circle of radius r is\frac{1}{r}.  (Proving that this is true of the formula above is a nice freshman calculus exercise.)
  • The circle that best approximates a curve at a point (x,y) is the circle that is tangent to the curve at  (x,y) and that has radius 1/k, where k is the curvature of the curve at (x,y) .   This circle is called the osculating circle at (x,y) . 

Curvature of the parabola

You can calculate that the curve of the parabola y=x^2 at x is given by

\kappa(x) = \frac{2}{\left(4 x^2+1\right)^{3/2}}

For example, the curvature at (0,1) is 2, at the point (1/2, 1/4)  it is  \scriptstyle 1/\sqrt{2}\:\approx\: 0.71, and at (1,1) it is about 0.18.  The radii of the osculating circles are  1/2,  \sqrt{2}, and 5.59 respectively.  For large numbers the curvature is nearly 0; for example, at (10, 100) the curvature is about .00025.  To the eye the parabola near (10,100)  looks like a straight line.

This graph shows the osculating circles at x = 0, 1/2 and 1:

threecircles

You can see animated osculating circles at the Wolfram Demonstration Project (click on “web preview”).  From that site you may download Mathematica Player for free, which allows you to operate the slidebars yourself. 

This graph shows the parabola and its curvature function.

   curvature21

Turning the wheel

If you think of the graph of the curve as a path and you imagine bicycling along the path, the size of the curvature corresponds to the specific angle to the right or left the front wheel must be turned to stay on the path. 

A circle has constant curvature, so to bike around a circle means keeping the front wheel at a constant angle. 

As you can see the curvature of the parabola goes up gradually as you move from a negative x– value to 0, and after that it goes down gradually.   So biking along that path from left to right means gradually turning your wheel to the left, and then at (0,0) you gradually turn it back closer to straight front. 

Notice that going faster or slower makes no difference to the angle you must turn the wheel (as long as you don’t skid).  The curvature at a point on the path depends on the path (which doesn’t move), not on the speed of your bicycle moving along the path. 

Another curve

You may have a seen a model electric train in action.   What I am about to say applies particular to cheap model trains.  They tend to have two kinds of track pieces, straight segments and segments of circles of fixed radius.  You could make a layout with these pieces that looks like this:

circleline1

When the train starts at the left, it goes along a straight track (curvature 0) until it reaches the point (0, 2), where it enters a stretch of constant curvature 1/2.   At (0, 2) the curvature jumps instantaneously from 0 to 1/2.   Of course, “instantaneous” does not exist in the physical world (at this scale — don’t start carrying on about quantum jumps, please).   Where the track starts to curve, the front wheels of the train are forced by the change in the track to suddenly jump from facing straight front to angling right by a fixed amount.  If you have the track on the floor and stand looking down at it, and the train is going pretty fast, you will notice that the front car jerks to the right as it enters the curve.

You can see this in action in this You-Tube movie at 15 seconds and 1:14 minutes.

Fancier model trains have track pieces with varying curvatures.  Look up “model train” on YouTube and you will see dozens of them.

If a highway were laid out like the graph above, and you were driving pretty fast, then at (0,2) you would have to turn your steering wheel suddenly to the right and you would probably swerve a little.  But you probably can’t find any highways like that.  In the 1960’s a Kentucky highway engineer told me that they knew better; they used French Curves with curvature that increases continuously from 0.  Nowadays highway engineers lay out highways using CAD systems that can calculate the track transition curves directly.

Send to Kindle
math, proof, representations, understanding math

Proofs without dry bones

3 Comments

I have discussed images, metaphors and proofs in math in two ways:

(A) A mathematical proof

A monk starts at dawn at the bottom of a mountain and goes up a path to the top, arriving there at dusk. The next morning at dawn he begins to go down the path, arriving at dusk at the place he started from on the previous day. Prove that there is a time of day at which he is at the same place on the path on both days.

Proof: Envision both events occurring on the same day, with a monk starting at the top and another starting at the bottom at the same time and doing the same thing the monk did on different days. They are on the same path, so they must meet each other. The time at which they meet is the time required.

This example comes from Fauconnier, Mappings in Thought and Language, Cambridge Univ. Press, 1997. I discuss it in the Handbook, pages 46 and 153. See the Wikipedia article on conceptual blending.

(B) Rigor and rigor mortis

The following is quoted from a previous post here. See also the discussion in abstractmath.

When we are trying to understand or explain math, we may use various kinds of images and metaphors about the subject matter to construct a colorful and rich representation of the mathematical objects and processes involved. I described some of these briefly here. They can involve thinking of abstract things moving and changing and affecting each other.

When we set out to prove some math statement, we go into what I have called “rigorous mode”. We feel that we have to forget some of the color and excitement of the rich view. We must think of math objects as inert and static. They don’t move or change over time and they don’t interact with other objects or the real world. In other words, pretend that all math objects are dead.

We don’t always go all the way into this rigorous mode, but if we use an image or metaphor in a proof and someone challenges us about it, we may rewrite that part to get rid of the colorful representation and replace it by a calculation or line of reasoning that refers to the math objects as if they were inert and static – dead.

I didn’t contradict myself.
I want to clear up some tension between these two ideas.

The argument in (A) is a genuine mathematical proof, just as it is written. It contains hidden assumptions (enthymemes), but all math proofs contain hidden assumptions. My remarks in (B) do not mean that a proof is not a proof until everything goes dead, but that when challenged you have to abandon some of the colorful and kinetic reasoning to make sure you have it right. (This is a standard mathematical technique (note 1).)

One of the hidden assumptions in (A) is that two monks walking the opposite way on the path over the same interval of time will meet each other. This is based on our physical experience. If someone questions this we have several ways to get more rigorous. One many mathematicians might think of is to model the path as a curve in space and consider two different parametrizations by the unit interval that go in opposite directions. This model can then appeal to the intermediate value theorem to assert that there is a point where the two parametrizations give the same value.

I suppose that argument goes all the way to the dead. In the original argument the monk is moving. But the parametrized curve just sits there. The parametrizations are sets of ordered pairs in R x (R x R x R). Nothing is moving. All is dry bones. Ezekiel has not done his thing yet.

This technique works, I think, because it allows classical logic to be correct. It is not correct in everyday life when things are moving and changing and time is passing.

Avoid models; axiomatize directly
But it certainly is not necessary to rigorize this argument by using parametrizations involving the real numbers. You could instead look at the situation of the monk and make some axioms the events being described. For example, you could presumably make axioms on locations on the path that treat the locations as intervals rather than as points.

The idea is to make axioms that state properties that intervals have but doesn’t say they are intervals. For example that there is a relation “higher than” between locations that must be reflexive and transitive but not antisymmetric. I have not done this, but I would propose that you could do this without recreating the classical real numbers by the axioms. (You would presumably be creating the intuitionistic real numbers.)

Of course, we commonly fall into using the real numbers because methods of modeling using real numbers have been worked out in great detail. Why start from scratch?

About the heading on this section: There is a sense in which “axiomatizing directly” is a way of creating a model. Nevertheless there is a distinction between these two approaches, but I am to confused to say anything about this right now.

First order logic.
It is commonly held that if you rigorize a proof enough you could get it all the way down to a proof in first order logic. You could do this in the case of the proof in (A) but there is a genuine problem in doing this that people don’t pay enough attention to.

The point is you replace the path and the monks by mathematical models (a curve in space) and their actions by parametrizations. The resulting argument calls on well known theorems in real analysis and I have no doubt can be turned into a strict first order logic argument. But the resulting argument is no longer about the monk on the path.

The argument in (A) involves our understanding of a possibly real physical situation along with a metaphorical transference in time of the two walks (a transference that takes place in our brain using techniques (conceptual blending) the brain uses every minute of every day). Changing over to using a mathematical model might get something wrong. Even if the argument using parametrized curves doesn’t have any important flaws (and I don’t believe it does) it is still transferring the argument from one situation to another.

Conclusion:
Mathematical arguments are still mathematical arguments whether they refer to mathematical objects or not. A mathematical argument can be challenged and tested by uncovering hidden assumptions and making them explicit as well as by transferring the argument to a classical mathematical situation.

Note 1. Did you ever hear anyone talking about rigor requiring making images and metaphors dead? This is indeed a standard mathematical technique but it is almost always suppressed, or more likely unnoticed. But I am not claiming to be the first one to reveal it to the world. Some of the members of Bourbaki talked this way. (I have lost the reference to this.)

They certainly killed more metaphors than most mathematicians.

Note 2. This discussion about rigor and dead things is itself a metaphor, so it involves a metametaphor. Metaphors always have something misleading about them. Metametaphorical statements have the potential of being far worse. For example, the notion that mathematics contains some kind of absolute truth is the result of bad metametaphorical thinking.

Send to Kindle
math, understanding math

Constraints on the Philosophy of Mathematics

Leave a comment

In a recent blog post I described a specific way in which neuroscience should constrain the philosophy of math. For example, many mathematicians who produce a new kind of mathematical object feel they have discovered something new, so they may believe that mathematical objects are created rather than eternally existing. But identifying something as newly created is presumably the result of a physical process in the brain. So the feeling that an object is new is only indirectly evidence that the object is new.  (Our pattern recognition devices work pretty well with respect to physical objects so that feeling is indeed indirect evidence.)

This constraint on philosophy is not based on any discovery that there really is a process in the brain devoted to recognizing new things. (Déjà vu is probably the result of the opposite process.) It’s just that neuroscience has uncovered very strong evidence that mental events like that are based on physical processes in the brain. Because of that work on other processes, if someone claims that recognizing newness is not based on a physical process in the brain, the burden of proof is on them.  In particular, they have to provide evidence that recognizing that a mathematical object is newly discovered says something about math other than what happened in your brain.

Of course, it will be worthwhile to investigate how the feeling of finding something new arises in the brain in connection with mathematical objects. Understanding the physical basis for how the brain does math has the potential of improving math education, although that may be years down the road.

Send to Kindle
language, math, representations, understanding math

Typical examples

2 Comments

There is a sense in which a robin is a typical bird and a penguin is not a typical bird. (See Reference 1.) Mathematical objects are like this and not like this.

A mathematical definition is simply a list of properties. If an object has all the properties, it is an example of the definition. If it doesn’t, it is not an example. So in some basic sense no example is any more typical than any other. This is in the sense of rigorous thinking described in the abstractmath chapter on images and metaphors.

In fact mathematicians are often strongly opinionated about examples: Some are typical, some are trivial, some are monstrous, some are surprising. A dihedral group is more nearly a typical finite group than a cyclic group. The real numbers on addition is not typical; it has very special properties. The monster group is the largest finite simple group; hardly typical. Its smallest faithful representation as complex matrices involves matrices that are of size 196,883. The monster group deserves its name. Nevertheless, the monster group is no more or less a group than the trivial group.

People communicating abstract math need to keep these two ideas in mind. In the rigorous sense, any group is just as much a group as any other. But when you think of an example of a group, you need to find one that is likely to provide some information. This depends on the problem. For some problems you might want to think of dihedral groups. For others, large abelian p-groups. The trivial group and the monster group are not usually good examples. This way of thinking is not merely a matter of psychology, but it probably can’t be made rigorous either. It is part of the way a mathematician thinks, and this aspect of doing math needs to be taught explicitly.

Reference 1. Lakoff, G. (1986), Women, Fire, and Dangerous Things. The University of Chicago Press. He calls typical examples “prototypical”.

Send to Kindle
abstractmath.org, language, language of math, math, representations, understanding math

Mathematical definitions

1 Comment

The definition of a concept in math has properties that are different from definitions in other subjects:

• Every correct statement about the concept follows logically from its definition.
• An example of the concept fits all the requirements of the definition (not just most of them).
• Every math object that fits all the requirements of the definition is an example of the concept.
• Mathematical definitions are crisp, not fuzzy.
• The definition gives a small amount of structural information and properties that are enough to determine the concept.
• Usually, much else is known about the concept besides what is in the definition.
• The info in the definition may not be the most important things to know about the concept.
• The same concept can have very different-looking definitions. It may be difficult to prove they give the same concept.
• Math texts use special wording to give definitions. Newcomers may not understand that the intent of an assertion is that it is a definition.

How many college math teachers ever explain these things?

I will expand on some of these concepts in future posts.

Send to Kindle
math, representations, understanding math

Math and the Modules of the Mind

6 Comments

I have written (references below) about the way we seem to think about math objects using our mind’s mechanisms for thinking about physical objects. What I want to do in this post is to establish a vocabulary for talking about these ideas that is carefully enough defined that what I say presupposes as little as possible about how our mind behaves. (But it does presuppose some things.) This is roughly like Gregor Mendel’s formulation of the laws of inheritance, which gave precise descriptions of how characteristics were inherited while saying nothing at all about the mechanism.

I will use module as a name for the systems in the mind that perform various tasks.

Examples of modules

a) We have an “I’ve seen this before module” that I talked about here.

b) When we see a table, our mind has a module that recognizes it as a table, a module that notes that it is nearby, and in particular a module that notes that it is a physical object. The physical-object module is connected to many other modules, including for example expectations of what we would feel if we touched it, and in particular connections to our language-producing module that has us talk about it in a certain way (a table, the table, my table, and so on.)

c) We also have a module for abstract objects. Abstract objects are discussed in detail in the math objects chapter of abstractmath.org. A schedule is an abstract object, and so is the month of November. They are not mathematical objects because they affect people and change over time. (More about this here.) For example, the statement “it is now November” is true sometimes and false sometimes. Abstract objects are also not abstractions, like “beauty” and “love” which are not thought of as objects.

d) We talk about numbers in some ways like we talk about physical objects. We say “3 is a number”. We say “I am thinking of the only even prime”. But if we point and say, “Look, there is a 3”, we know that we have shifted ground and are talking about, not the number 3, but about a physical representation of the number 3. That’s because numbers trigger our abstract object module and our math object module, but not our physical object module. (Back and fill time: if you are not a mathematician, your mind may not have a math object module. People are not all the same.)

More about modules

My first choice for a name for these systems would have been object, as in object-oriented programming, but this discussion has too many things called objects already. Now let’s clear up some possible misconceptions:

e) I am talking about a module of the mind. My best guess would be that the mind is a function of the brain and its relationship with the world, but I am not presuppposing that. Whatever the mind is, it obviously has a system for recognizing that something is a physical object or a color or a thought or whatever. (Not all the modules are recognizers; some of them initiate actions or feelings.)

f) It seems likely that each module is a neuron together with its connections to other neurons, with some connections stronger than others (our concepts are fuzzy, not Boolean). But maybe a module is many neurons working together. Or maybe it is like a module in a computer program, that is instantiated anew each time it is called, so that a module does not have a fixed place in the brain. But it doesn’t matter. A module is whatever it is that carries out a particular function. Something has to carry out such functions.

Math objects

The modules in a mathematician’s mind that deal with math objects use some of the same machinery that the mind uses for physical objects.

g) You can do things to them. You can add two numbers. You can evaluate a function at an input. You can take the derivative of some functions.

h) You can discover properties of some kinds of math objects. (Every differentiable function is continuous.)

i) Names of some math objects are treated as proper nouns (such as “42”) and others as common nouns (such as “a prime”.)

I maintain that these phenomena are evidence that the systems in your mind for thinking about physical objects are sometimes useful for thinking about math objects.

Different ways of thinking about math objects.

j) You can construct a mathematical object that is new to you. You may feel that you invented it, that it didn’t exist before you created it. That’s your I just created this module acting. If you feel this way, you may think math is constantly evolving.

k) Many mathematicians feel that math objects are all already there. That’s a module that recognizes that math objects don't come into or go out of existence.

l) When you are trying to understand math objects you use all sorts of physical representations (graphs, diagrams) and mental representations (metaphors, images). You say things like, “This cubic curve goes up to positive infinity in the negative direction” and “This function vanishes at 2” and “Think of a Möbius strip as the unit square with two parallel sides identified in the reverse direction.”

m) When you are trying to prove something about math objects mathematicians generally think of math objects as eternal and inert (not affecting anything else). For example, you replace “the slope of the secant gets closer and closer to the slope of the tangent” by an epsilon-delta argument in which everything you talk about is treated as if it is unchanging and permanent. (See my discussion of the rigorous view.)

Consequences

When you have a feeling of déjà vu, it is because something has triggered your “I have seen this before” module (see (a)). It does not mean you have seen it before.

When you say “the number 3” is odd, that is a convenient way of talking about it (see (d) above), but it doesn’t mean that there is really only one number three.

If you say the function x^2 takes 3 to 9 it doesn’t have physical consequences like “Take me to the bank” might have. You are using your transport module but in a pretend way (you are using the pretend module!).

When you think you have constructed a new math object (see (j)), your mental modules leave you feeling that the object didn’t exist before. When you think you have discovered a new math object (see (k)), your modules leave you feeling that it did exist before. Neither of those feelings say anything about reality, and you can even have both feelings at the same time.

When you think about math objects as eternal and inert (see (m)) you are using your eternal and inert modules in a pretend way. This does not constitute an assertion that they are eternal and inert.

Is this philosophy?

My descriptions of how we think about math are testable claims about the behavior of our mind, expressed in terms of modules whose behavior I (partially) specify but whose nature I don’t specify. Just as Mendel’s Laws turned out to be explained by the real behavior of chromosomes under meiosis, the phenomena I describe may someday turn out to be explained by whatever instantiation the modules actually have – except for those phenomena that I have described wrongly, of course – that is what “testable” means!

So what I am doing is science, not philosophy, right?

Now my metaphor-producing module presents the familiar picture of philosophy and science as being adjacent countries, with science intermittently taking over pieces of philosophy’s territory…

Links to my other articles in this thread

Math objects in abstractmath.org
Mathematical objects are “out there”?
Neurons and math
A scientific view of mathematics (has many references to what other people have said about math objects)
Constructivism and Platonism

Send to Kindle
abstractmath.org, exposition, Handbook, language of math, math, proof, understanding math

Abstractmath.org after four years

Leave a comment

I have been working on the abstractmath website for about four years now (with time off for three major operations). Much has been written, but there are still lots of stubs that need to be filled in. Also much of it needs editing for stylistic uniformity, and for filling in details and providing more examples in some hastily written sections that read like outlines. Not to mention correcting errors, which seem to multiply when I am not looking. The website consists of four main parts and some ancillary chapters. I will go into more detail about some of the parts in later articles.

The languages of math. This is a description of mathematical English and the symbolic language of math (which are two different languages!) with an emphasis on the problems they cause people new to abstract math (roughly, math after calculus). At this point, I have completed a fairly thorough edit of the whole chapter that makes it almost presentable. Start with the Introduction.

Proofs. Mathematical proofs are a central problem for abstract math newbies. People interested in abstract math must learn to read and understand proofs. A proof is narrated in mathematical English. A proof has a logical structure. The reader must extract the logical structure from the narrative form. The chapter on proofs gives examples of proofs and discusses the logical structure and its relationship with the narration. The introduction to the chapter on proofs tells more about it.

Understanding math. There are certain barriers to understanding math that are difficult to get over. Mathematicians, math educators and philosophers work on various aspects of these problems and this chapter draws on their work and my own observations as a mathematician and a teacher.

All true statements about a math object must follow from the definition. That sounds clear enough. But in fact there are subtleties about definitions teachers may not tell students about because they are not aware of them themselves. For example, a definition can really mislead you about how to think about a math object.

The section on math objects breaks new ground (in my opinion) about how to think about them. I also discuss representations and models and images and metaphors (which I think is especially important), and in shorter articles about other topics such as abstraction and pattern recognition.

Doing math. This chapter points out useful behaviors and dysfunctional behaviors in doing math, with concrete examples. Beginners need to be told that when proving an elementary theorem they need to rewrite what is to be proved according to the definitions. Were you ever told that? (If you went to a Jesuit high school, you probably were.) Beginners need to be told that they should not try the same computational trick over and over even though it doesn’t work. That they need to look at examples. That they need to zoom in and out, looking at a detail and then the big picture. We need someone to make movies illustrating these things.

These other articles are outside the main organization:
Topic articles. Sets, real numbers, functions, and so on. In each case I talk just a bit about the topic to get the newbie over the initial hump.
Diagnostic examples. Examples chosen to evoke a misunderstanding, with a link to where it is explained. This needs to be greatly expanded.
Attitudes. This explains my point of view in doing abstractmath.org. I expect to rewrite it.

Send to Kindle
math, understanding math

Constructivism and Platonism — a Third Way

5 Comments

This is my response to Phil Wilson's article on Constructivism in a recent Plus Magazine, in particular to the following paragraph (but please read the whole article!) where he talks about

'…how firmly entrenched is the realist view that mathematical objects exist independently of the human mind, "out there" somewhere just waiting to be discovered, even in our every day conception of objects as fundamental as the real numbers. Intuitionism is radically antirealist: antirealist in that it claims mathematical objects only come into existence once they are constructed by a human mind (a sad quirk of language that this is called "anti"realist), and radical since it seeks to recast all of mathematics in this light.'

We don't have to choose between the view that mathematical objects exist independently of the human mind and the view that they only come into existence once they are constructed by a human mind. There is a third approach: We think about mathematical objects as if they exist independently of the human mind. In particular, mathematicians have gotten away with pretending that all the digits of a real number exist all at once and proving theorems such as trichotomy based on that view, without running into contradictions. The justification is just that: it works.

This approach has the advantage that our brain has a whole system of thinking about physical objects. We use this system to think about other things such as Sherlock Holmes and pi and appointment schedules and it works quite well. It doesn't work perfectly: for example, physical objects change over time and affect each other, whereas we must think of mathematical objects as eternal and inert if our proof techniques are going to work properly. Indeed, it is thinking of the decimal digits of pi (for example) as "going toward infinity" that gets students into trouble with limits.

Even so, objectification, if that is the right word, has worked very well for mathematicians and we don't need to give it up, nor do we need to be Platonists — we need only act as if we are Platonists.

I wrote about this in several places:

A scientific view of mathematics

Rigor and rigor mortis

Rich and rigorous

 

Send to Kindle