Conceptual blending

2012/06/18 — SixWingedSeraph

This post uses MathJax.  If you see formulas in unrendered TeX, try refreshing the screen.

A conceptual blend is a structure in your brain that connects two concepts by associating part of one with part of another.  Conceptual blending is a major tool used by our brain to understand the world.

The concept of conceptual blend includes special cases, such as representations, images and conceptual metaphors, that math educators have used for years to understand how mathematics is communicated and how it is learned.  The Wikipedia article is a good starting place for understanding conceptual blending. 

In this post I will illustrate some of the ways conceptual blending is used to understand a function of the sort you meet with in freshman calculus.  I omit the connections with programs, which I will discuss in a separate post.

A particular function

Consider the function $h(t)=4-(t-2)^2$. You may think of this function in many ways.

FORMULA:

$h(t)$ is defined by the formula $4-(t-2)^2$.

  • The formula encapsulates a particular computation of the value of $h$ at a given value $t$.
  • The formula defines the function, which is a stronger statement than saying it represents the function.
  • The formula is in standard algebraic notation. (See Note 1)
  • To use the formula requires one of these:
    • Understand and use the rules of algebra
    • Use a calculator
    • Use an algebraic programming language. 
  • Other formulas could be used, for example $4t-t^2$.
    • That formula encapsulates a different computation of the value of $h$.

TREE: 

$h(t)$ is also defined by this tree (right).
  • The tree makes explicit the computation needed to evaluate the function.
  • The form of the tree is based on a convention, almost universal in computing science, that the last operation performed (the root) is placed at the top and that evaluation is done from bottom to top.
  • Both formula and tree require knowledge of conventions.
  • The blending of formula and tree matches some of the symbols in the formula with nodes in the tree, but the parentheses do not appear in the tree because they are not necessary by the bottom-up convention.
  • Other formulas correspond to other trees.  In other words, conceptually, each tree captures not only everything about the function, but everything about a particular computation of the function.
  • More about trees in these posts:

GRAPH:

$h(t)$ is represented by its graph (right). (See note 2.)

  • This is the graph as visual image, not the graph as a set of ordered pairs.
  • The blending of graph and formula associates each point on the (blue) graph with the value of the formula at the number on the x-axis directly underneath the point.
  • In contrast to the formula, the graph does not define the function because it is a physical picture that is only approximate.
  • But the formula does represent the function.  (This is "represents" in the sense of cognitive psychology, but not in the mathematical sense.)
  • The blending requires familiarity with the conventions concerning graphs of functions. 
  • It sets into operation the vision machinery of your brain, which is remarkably elaborate and powerful.
    • Your visual machinery allows you to see instantly that the maximum of the curve occurs at about $t=2$. 
  • The blending leaves out many things.
    • For one, the graph does not show the whole function.  (That's another reason why the graph does not define the function.)
    • Nor does it make it obvious that the rest of the graph goes off to negative infinity in both directions, whereas that formula does make that obvious (if you understand algebraic notation).  

GEOMETRIC

The graph of $h(t)$ is the parabola with vertex $(2,4)$, directrix $x=2$, and focus $(2,\frac{3}{4})$. 

  • The blending with the graph makes the parabola identical with the graph.
  • This tells you immediately (if you know enough about parabolas!) that the maximum is at $(2,4)$ (because the directrix is vertical).
  • Knowing where the focus and directrix are enables you to mechanically construct a drawing of the parabola using a pins, string, T-square and pencil.  (In the age of computers, do you care?)

HEIGHT:

$h(t)$ gives the height of a certain projectile going straight up and down over time.

  • The blending of height and graph lets you see instantly (using your visual machinery) how high the projectile goes. 
  • The blending of formula and height allows you to determing the projectile's velocity at any point by taking the derivative of the function.
  • A student may easily be confused into thinking that the path of the projectile is a parabola like the graph shown.  Such a student has misunderstood the blending.

KINETIC:

You may understand $h(t)$ kinetically in various ways.

  • You can visualize moving along the graph from left to right, going, reaching the maximum, then starting down.
    • This calls on your experience of going over a hill. 
    • You are feeling this with the help of mirror neurons.
  • As you imagine traversing the graph, you feel it getting less and less steep until it is briefly level at the maximum, then it gets steeper and steeper going down.
    • This gives you a physical understanding of how the derivative represents the slope.
    • You may have seen teachers swooping with their hand up one side and down the other to illustrate this.
  • You can kinetically blend the movement of the projectile (see height above) with the graph of the function.
    • As it goes up (with $t$ increasing) the projectile starts fast but begins to slow down.
    • Then it is briefly stationery at $t=2$ and then starts to go down.
    • You can associate these feelings with riding in an elevator.
      • Yes, the elevator is not a projectile, so this blending is inaccurate in detail.
    • This gives you a kinetic understanding of how the derivative gives the velocity and the second derivative gives the acceleration.

OBJECT:

The function $h(t)$ is a mathematical object.

  • Usually the mental picture of function-as-object consists of thinking of the function as a set of ordered pairs $\Gamma(h):=\{(t,4-(t-2)^2)|t\in\mathbb{R}\}$. 
  • Sometimes you have to specify domain and codomain, but not usually in calculus problems, where conventions tell you they are both the set of real numbers.
  • The blend object and graph identifies each point on the graph with an element of $\Gamma(h)$.
  • When you give a formal proof, you usually revert to a dry-bones mode and think of math objects as inert and timeless, so that the proof does not mention change or causation.
    • The mathematical object $h(t)$ is a particular set of ordered pairs. 
    • It just sits there.
    • When reasoning about something like this, implication statements work like they are supposed to in math: no causation, just picking apart a bunch of dead things. (See Note 3).
    • I did not say that math objects are inert and timeless, I said you think of them that way.  This post is not about Platonism or formalism. What math objects "really are" is irrelevant to understanding understanding math [sic].

DEFINITION

A definition of the concept of function provides a way of thinking about the function.

  • One definition is simply to specify a mathematical object corresponding to a function: A set of ordered pairs satisfying the property that no two distinct ordered pairs have the same second coordinate, along with a specification of the codomain if that is necessary.
  • A concept can have many different definitions.
    • A group is usually defined as a set with a binary operation, an inverse operation, and an identity with specific properties.  But it can be defined as a set with a ternary operation, as well.
    • A partition of a set is a set of subsets of a set with certain properties. An equivalence relation is a relation on a set with certain properties.  But a partition is an equivalence relation and an equivalence relation is a partition.  You have just picked different primitives to spell out the definition. 
    • If you are a beginner at doing proofs, you may focus on the particular primitive objects in the definition to the exclusion of other objects and properties that may be more important for your current purposes.
      • For example, the definition of $h(t)$ does not mention continuity, differentiability, parabola, and other such things.
      • The definition of group doesn't mention that it has linear representations.

SPECIFICATION

A function can be given as a specification, such as this:

If $t$ is a real number, then $h(t)$ is a real number, whose value is obtained by subtracting $2$ from $t$, squaring the result, and then subtracting that result from $4$.

  • This tells you everything you need to know to use the function $h$.
  • It does not tell you what it is as a mathematical object: It is only a description of how to use the notation $h(t)$.

Notes

1. Formulas can be give in other notations, in particular Polish and Reverse Polish notation. Some forms of these notations don't need parentheses.

2. There are various ways to give a pictorial image of the function.  The usual way to do this is presenting the graph as shown above.  But you can also show its cograph and its endograph, which are other ways of representing a function pictorially.  They  are particularly useful for finite and discrete functions. You can find lots of detail in these posts and Mathematica notebooks:

3. See How to understand conditionals in the abstractmath article on conditionals.

References

  1. Conceptual blending (Wikipedia)
  2. Conceptual metaphors (Wikipedia)
  3. Definitions (abstractmath)
  4. Embodied cognition (Wikipedia)
  5. Handbook of mathematical discourse (see articles on conceptual blend, mental representation, representation, and metaphor)
  6. Images and Metaphors (article in abstractmath)
  7. Links to G&G posts on representations
  8. Metaphors in Computing Science (previous post)
  9. Mirror neurons (Wikipedia)
  10. Representations and models (article in abstractmath)
  11. Representations II: dry bones (article in abstractmath)
  12. The transition to formal thinking in mathematics, David Tall, 2010
  13. What is the object of the encapsulation of a process? Tall et al., 2000.

 

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Stances

2009/04/03 — Charles Wells

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.

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Constraints on the Philosophy of Mathematics

2009/03/18 — Charles Wells

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.

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Math and the Modules of the Mind

2009/01/30 — Charles Wells

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

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