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Endograph and cograph of real functions

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This post is covered by the Creative Commons Attribution – ShareAlike 3.0 License, which means you may use, adapt and distribute the work provided you follow the requirements of the license.

Introduction

In the article Functions: Images and Metaphors in abstractmath I list a bunch of different images or metaphors for thinking about functions. Some of these metaphors have realizations in pictures, such as a graph or a surface shown by level curves. Others have typographical representations, as formulas, algorithms or flowcharts (which are also pictorial). There are kinetic metaphors — the graph of {y=x^2} swoops up to the right.

Many of these same metaphors have realizations in actual mathematical representations.

Two images (mentioned only briefly in the abstractmath article) are the cograph and the endograph of a real function of one variable. Both of these are visualizations that correspond to mathematical representations. These representations have been used occasionally in texts, but are not used as much as the usual graph of a continuous function. I think they would be useful in teaching and perhaps even sometimes in research.

A rough and unfinished Mathematica notebook is available that contains code that generate graphs and cographs of real-valued functions. I used it to generate most of the examples in this post, and it contains many other examples. (Note [1].)

The endograph of a function

In principle, the endograph (Note [2]) of a function {f} has a dot for each element of the domain and of the codomain, and an arrow from {x} to {f(x)} for each {x} in the domain. For example, this is the endograph of the function {n\mapsto n^2+1 \pmod 11} from the set {\{0,1,\ldots,10\}} to itself:


“In principle” means that the entire endograph can be shown only for small finite functions. This is analogous to the way calculus books refer to a graph as “the graph of the squaring function” when in fact the infinite tails are cut off.

Real endographs

I expect to discuss finite endographs in another post. Here I will concentrate on endographs of continuous functions with domain and codomain that are connected subsets of the real numbers. I believe that they could be used to good effect in teaching math at the college level.

Here is the endograph of the function {y=x^2} on the reals:

I have displayed this endograph with the real line drawn in the usual way, with tick marks showing the location of the points on the part shown.

The distance function on the reals gives us a way of interpreting the spacing and location of the arrowheads. This means that information can be gleaned from the graph even though only a finite number of arrows are shown. For example you see immediately that the function has only nonnegative values and that its increase grows with {x}.(See note [3]).

I think it would be useful to show students endographs such as this and ask them specific questions about why the arrows do what they do.

For the one shown, you could ask these questions, probably for class discussion rather that on homework.

  • Explain why most of the arrows go to the right. (They go left only between 0 and 1 — and this graph has such a coarse point selection that it shows only two arrows doing that!)
  • Why do the arrows cross over each other? (Tricky question — they wouldn’t cross over if you drew the arrows with negative input below the line instead of above.)
  • What does it say about the function that every arrowhead except two has two curves going into it?

Real Cographs

The cograph (Note [4] of a real function has an arrow from input to output just as the endograph does, but the graph represents the domain and codomain as their disjoint union. In this post the domain is a horizontal representation of the real line and the codomain is another such representation below the domain. You may also represent them in other configurations (Note [5]).

Here is the cograph representation of the function {y=x^2}. Compare it with the endograph representation above.

Besides the question of most arrows going to the right, you could also ask what is the envelope curve on the left.

More examples

Absolute value function

Arctangent function

Notes

[1] This website contains other notebooks you might find useful. They are in Mathematica .nb, .nbp, or .cdf formats, and can be read, evaluated and modified if you have Mathematica 8.0. They can also be made to appear in your browser with Wolfram CDF Player, downloadable free from Wolfram site. The CDF player allows you to operate any interactive demos contained in the file, but you can’t evaluate or modify the file without Mathematica.

The notebooks are mostly raw code with few comments. They are covered by the Creative Commons Attribution – ShareAlike 3.0 License, which means you may use, adapt and distribute the code following the requirements of the license. I am making the files available because I doubt that I will refine them into respectable CDF files any time soon.

[2] I call them “endographs” to avoid confusion with the usual graphs of functions — — drawings of (some of) the set of ordered pairs {x,f(x)} of the function.

[3] This is in contrast to a function defined on a discrete set, where the elements of the domain and codomain can be arranged in any old way. Then the significance of the resulting arrangement of the arrows lies entirely in which two dots they connect. Even then, some things can be seen immediately: Whether the function is a cycle, permutation, an involution, idempotent, and so on.

Of course, the placement of the arrows may tell you more if the finite sets are ordered in a natural way, as for example a function on the integers modulo some integer.

[4] The text [1] uses the cograph representation extensively. The word “cograph” is being used with its standard meaning in category theory. It is used by graph theorists with an entirely different meaning.

[5] It would also be possible to show the domain codomain in the usual {x-y} plane arrangement, with the domain the {x} axis and the codomain the {y} axis. I have not written the code for this yet.

References

[1] Sets for Mathematics, by F. William Lawvere and Robert Rosebrugh. Cambridge University Press, 2003.

[2] Martin Flashman’s website contains many exampls of cographs of functions, which he calls mapping diagrams.

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Mental, Physical and Mathematical Representations

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For a given mathematical object, a mathematician may have:

  • A mental representation of the object. This can be a metaphor, a mental image, or a kinesthetic understanding of the object.
  • A physical representation of the object. This may be a (physical) picture or drawing or three-dimensional model of the object.
  • A mathematical definition and one or more mathematical representations for the object. Such a representation is itself a mathematical object.

The boldface things in this list are related to each other in lots of ways, and they are fuzzy and overlap and don’t include every phenomenon connected with a math object.

I have written about these things ([1], [2], [3], [4]). So have lots of other people. In this post I summarize these ideas. I expect to write about particular examples later on and will use this as a reference.

Two Examples

The following examples point out a few of the relationships between the ideas in boldface above. There is much more to understand.

Function as black box

The idea that a function is a black box or machine with input and output is a metaphor for a function.

A is a metaphor for B means that A and B are cognitively pasted together in such a way that the behavior of A is in many ways like the behavior of B. Such a thing is both useful and dangerous, dangerous because there will be ways in which A behaves that suggest inappropriate ideas about B.

The function as machine is a good metaphor: for example functional composition involves connecting the output of one machine to the input of another, and the inverse function is like running the machine backward.

The function as machine is a bad metaphor: For example, it is wrong to think you could build a machine to calculate any given function exactly. But you can still imagine such a machine, given by a specification (it outputs the value of the function at a given input) and then, in your imagination, connecting the input of one to the output of another must perforce calculate the composite of the corresponding functions.

Like any metaphor, this is a mental representation. That means the metaphor has a physical instantiation in your brain. So a metaphor has a physical representation.

Different people won’t have quite the same concept of a particular metaphor. So a metaphor will have lots of slightly different physical representations, but mathematicians form a community, and communication between mathematicians fine-tunes the different physical instantiations so that they correspond more closely to each other. This is the sense in which mathematical objects have a shared existence in a community as Reuben Hersh has suggested.

A function is a mathematical object, which can be rigorously specified as a set of ordered pairs together with a domain and a codomain. There is a cognitive relationship between the concepts of function as math object and function as black box with input and output.

Triangle

A triangle can be drawn, or created on a computer and a physical image printed out. You may also have a mental image of the triangle.

The physical and the mental images are not the same thing, but they are definitely related. The relationship is mediated by the neuronal circuitry behind your retinas, which performs a highly sophisticated transformation of the pixels on your retina into an organized physical structure in your brain, connected to various other neurons.

This circuitry exists because it helps us get a useful understanding of the world through our eyes. So a picture of a triangle takes advantage of pre-existing neuron structure to generate a useful mental representation that helps us understand and prove things about triangles.

This mental representation also lives in a community of mathematician. Like any community, it has subgroups with “dialects” — varying understanding of representation.

For example, a mathematician who looks at the triangle below sees a triangle that looks like a right triangle. A student sees a triangle that is a right triangle.

This is “sees” in the sense of what their brain reports after all that processing. The mathematician’s brain connects the “triangle I am seeing” module (in their brain) to the “looks like a right triangle” module, but does not connect it to the “is a right triangle” module because they don’t see any statement in the surrounding text that it is a right triangle. The student, on the other hand, fallaciously makes the connection to “is a right triangle” directly.

In some sense, a student who does not make that connection directly is already a mathematician.

A triangle also exists as a mathematical object in your and my brain. It is described by a formal mathematical definition. The pictures of triangles you see above do not fit this definition. For one thing, the line segments in the pictures have thickness. But the pictures trigger a reaction in your neurons that causes your brain to cognitively paste together the line segments in the drawing to the segments required by the formal definition. This is a kind of metaphor of concrete-to-abstract that connects drawings to math objects that mathematicians use all the time.

Note that this “concrete-to-abstract metaphor” itself has a physical existence in your brain.  It drops, for example, the property of thickness that the line segments in the drawing have when matching them (in the metaphor) with the line segments in the corresponding abstract triangle.  On the other hand, it preserves the sense the all three angles in the triangle are acute.  The abstract mathematical concept of triangle (the generic triangle) has no requirement on the angles except that they add up to pi.

Summary

The discussions above describe a few of the complex and subtle relationships that exist between

  • Mental representations of math objects
  • Physical representations of math objects
  • Formally defined math objects and their formally defined representations.

I have purported to discuss how mathematics is understood (especially in connection with language) in several articles and a book but only a few of the relationships I just described are mentioned in any of those articles. Perhaps one or two things I said caused you to react: “Actually, that’s obviously true but I never thought of it before”. (Much the way I had mathematicians in the ’60’s tell me, “I see what you mean that addition is a function of two variables, but I never thought of it that way before”.) (I was a brash category theorist wannabe then.)

A lot of research has been done on understanding math, and some research has been done on mathematical discourse. But what has been done has merely exposed the fin of the shark.

References

[1] Images and metaphors (in abstractmath).

[2] Representations and Models (in abstractmath).

[3] Mathematical Concepts (previous blog).

[4] Mental Representations in Math (previous blog).

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Function as map

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This is a first draft of an article to eventually appear in abstractmath.

Images and metaphors

To explain a math concept, you need to explain how mathematicians think about the concept. This is what in abstractmath I call the images and metaphors carried by the concept. Of course you have to give the precise definition of the concept and basic theorems about it. But without the images and metaphors most students, not to mention mathematicians from a different field, will find it hard to prove much more than some immediate consequences of the definition. Nor will they have much sense of the place of the concept in math and applications.

Teachers will often explain the images and metaphors with handwaving and pictures in a fairly vague way. That is good to start with, but it’s important to get more precise about the images and metaphors. That’s because images and metaphors are often not quite a good fit for the concept — they may suggest things that are false and not suggest things that are true. For example, if a set is a container, why isn’t the element-of relation transitive? (A coin in a coinpurse in your pocket is a coin in your pocket.)

“A metaphor is a useful way to think about something, but it is not the same thing as the same thing.” (I think I stole that from the Economist.) Here, I am going to get precise with the notion that a function is a map. I am acting like a mathematician in “getting precise”, but I am getting precise about a metaphor, not about a mathematical object.

A function is a map

A map (ordinary paper map) of Minnesota has the property that each point on the paper represents a point in the state of Minnesota. This map can be represented as a mathematical function from a subset of a 2-sphere to {{\mathbb R}^2}. The function is a mathematical idealization of the relation between the state and the piece of paper, analogous to the mathematical description of the flight of a rocket ship as a function from {{\mathbb R}} to {{\mathbb R}^3}.

The Minnesota map-as-function is probably continuous and differentiable, and as is well known it can be angle preserving or area preserving but not both.

So you can say there is a point on the paper that represents the location of the statue of Paul Bunyan in Bemidji. There is a set of points that represents the part of the Mississippi River that lies in Minnesota. And so on.

A function has an image. If you think about it you will realize that the image is just a certain portion of the piece of paper. Knowing that a particular point on the paper is in the image of the function is not the information contained in what we call “this map of Minnesota”.

This yields what I consider a basic insight about function-as-map:  The map contains the information about the preimage of each point on the paper map. So:

The map in the sense of a “map of Minnesota” is represented by the whole function, not merely by the image.

I think that is the essence of the metaphor that a function is a map. And I don’t think newbies in abstractmath always understand that relationship.

A morphism is a map

The preceding discussion doesn’t really represent how we think of a paper map of Minnesota. We don’t think in terms of points at all. What we see are marks on the map showing where some particular things are. If it is a road map it has marks showing a lot of roads, a lot of towns, and maybe county boundaries. If it is a topographical map it will show level curves showing elevation. So a paper map of a state should be represented by a structure preserving map, a morphism. Road maps preserve some structure, topographical maps preserve other structure.

The things we call “maps” in math are usually morphisms. For example, you could say that every simple closed curve in the plane is an equivalence class of maps from the unit circle to the plane. Here equivalence class meaning forget the parametrization.

The very fact that I have to mention forgetting the parametrization is that the commonest mathematical way to talk about morphisms is as point-to-point maps with certain properties. But we think about a simple closed curve in the plane as just a distorted circle. The point-to-point correspondence doesn’t matter. So this example is really talking about a morphism as a shape-preserving map. Mathematicians introduced points into talking about preserving shapes in the nineteenth century and we are so used to doing that that we think we have to have points for all maps.

Not that points aren’t useful. But I am analyzing the metaphor here, not the technical side of the math.

Groups are functors

People who don’t do category theory think the idea of a mathematical structure as a functor is weird. From the point of view of the preceding discussion, a particular group is a functor from the generic group to some category. (The target category is Set if the group is discrete, Top if it is a topological group, and so on.)

The generic group is a group in a category called its theory or sketch that is just big enough to let it be a group. If the theory is the category with finite products that is just big enough then it is the Lawvere theory of the group. If it is a topos that is just big enough then it is the classifying topos of groups. The theory in this sense is equivalent to some theory in the sense of string-based logic, for example the signature-with-axioms (equational theory) or the first order theory of groups. Johnstone’s Elephant book is the best place to find the translation between these ideas.

A particular group is represented by a finite-limit-preserving functor on the algebraic theory, or by a logical functor on the classifying topos, and so on; constructions which bring with them the right concept of group homomorphisms as well (they will be any natural transformations).

The way we talk about groups mimics the way we talk about maps. We look at the symmetric group on five letters and say its multiplication is noncommutative. “Its multiplication” tells us that when we talk about this group we are talking about the functor, not just the values of the functor on objects. We use the same symbols of juxtaposition for multiplication in any group, “{1}” or “{e}” for the identity, “{a^{-1}}” for the inverse of {a}, and so on. That is because we are really talking about the multiplication, identity and inverse function in the generic group — they really are the same for all groups. That is because a group is not its underlying set, it is a functor. Just like the map of Minnesota “is” the whole function from the state to the paper, not just the image of the function.

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Mental Representations in Math

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This post is part of the abstractmath article on images and metaphors.  I have had some new insights into the subject of mental representations and have incorporated them in this rewritten version (which omits some examples).  I would welcome comments.

Mathematicians who work with a particular kind of mathematical object have mental representations of that type of object that help them understand it.  These mental representations come in various forms:

  • Visual images,  for example of what a right triangle looks like.
  • Notation, for example visualizing the square root of 2 by the symbol “\sqrt{2}“.  Of course, in a sense notation is also a physical representation of the number.  An important fact:  A mathematical object may be referred to by many different notations. There are examples here and here. (If you think deeply about the role notation plays in your head and on paper you can easily get a headache.)
  • Kinetic understanding, for example thinking of the value of a function as approaching a limit by imagining yourself moving along the graph of the function.
  • Metaphorical understanding, for example thinking of a function such as  as a machine that turns one number into another: for example, when you put in 3 out comes 9.   See also literalism and this post on Gyre&Gimble.

Example

Consider the function h(t)=25-(t-5)^2.   The chapter on images and metaphors for functions describes many ways to think about functions.  A few of them are considered here.

Visual images You can picture this function in terms of its graph, which is a parabola.   You can think of it more physically, as like the Gateway Arch.  The graph visualization suggests that the function has a single maximum point that appears to occur at t = 5.

I personally use visual placement to remember relationships between abstract objects, as well.  For example, if I think of three groups, two of which are isomorphic (for example C_2 and \text{Alt}_3.), I picture them as in different places with a connection between the two isomorphic ones.  I know of no research on this.

Notation You can think of the function as its formula .  The formula tells you that its graph will be a parabola (if you know that quadratics give parabolas) and it tells you instantly without calculus that its maximum will be at (see ratchet effect).

Another formula for the same function is -t^2+10t.   The formula is only a representation of the function.  It is not the same thing as the function.  The functions h(t) and k(t) defined on the real numbers  by h(t)=25-(t-5)^2 and k(t)=-t^2+10t are the same function; in other words, h = k.

Kinetic The function h(t)  could model the height over time of a physical object, perhaps a ball shot vertically upwards on a planet with no atmosphere. You could think of the ball starting at time t = 0 at elevation 0, reaching an elevation of (for example) 16 units at time t = 2, and landing at t = 10.  You are imagining a physical event continuing over time, not just as a picture but as a feeling of going up and down (see mirror neuron).  This feeling of the ball going up and down is attached in your brain to your understanding of the function h(t).

Although h(t) models the height of the ball, it is not the same thing as the height of the ball. A mathematical object may have a relationship in our mind to physical processes or situations but is distinct from them.

According to this report, kinetic understanding can also help with learning math that does not involve pictures.  I know that when I think of evaluating the function  at 3, I visualize 3 moving into the x slot and then the formula  transforming itself into 10.  I remember doing this even before I had ever heard of the Transformers.

Metaphor One metaphor for functions is that it is a machine that turns one number into another.  For example, the function h(t)  turns 0 into 0 (which is therefore a fixed point) and 5 into 25.  It also turns 10 into 0, so once you have the output you can’t necessarily tell what the input was (the function is not injective).

More examples

  • ¨ “Continuous functions don’t have gaps in the graph“. This is a visual image.
  • ¨ You may think of the real numbers as lying along a straight line (the real line) that extends infinitely far in both directions. This is both visual and a metaphor (a real number “is” a place on the real line).
  • ¨ You can think of the set containing 1, 3 and 5 and nothing else in terms of its list notation {1, 3, 5}. But remember that {5, 1,3} is the same set. In other words the list notation has irrelevant features – the order in which the elements are listed in this case.
  • The interior of a closed curve or a sphere is called that because it is like the interior in the everyday sense of a bucket or a house. Note The boundary of a real-life container such as a bucket has thickness, in contrast to the boundary of a closed curve or a sphere. This observation illustrates my description of a metaphor as identifying part of one situation with part of another. One aspect is emphasized; another aspect, where they may differ, is ignored.

Uses of mental representations

Integers and metaphors make up what is arguably the most important part of the mathematician’s understanding of the concept.

  • Mental representations of mathematical objects using metaphors and images are necessary for understanding and communicating about them (especially with types of objects that are new to us).
  • They are necessary for seeing how the theory can be applied. 
  • They are useful for coming up with proofs.

Many representations

Different mental representations of the same kind of object help you understand different aspects of the object.

Every important mathematical object has many representations and skilled mathematicians generally have several of them in mind at once.

New concepts and old ones

We especially depend on metaphors and images to understand a math concept that is new to us.  But if we work with it for awhile, finding lots of examples, and eventually proving theorems and providing counterexamples to conjectures, we begin to understand the concept in its own terms and the images and metaphors tend to fade away from our awareness…

Then, when someone asks us about this concept that we are now experts with, we trundle out our old images and metaphors – and are often surprised at how difficult and misleading our listener finds them!

Some mathematicians retreat from images and metaphors because of this and refuse to do more than state the definition and some theorems about the concept.   They are wrong to do this. That behavior encourages the attitude of many people that

  • mathematicians can’t explain things
  • math concepts are incomprehensible or bizarre
  • you have to have a mathematical mind to understand math

All three of these statements are half-truths. There is no doubt that a lot of abstract math is hard to understand, but understanding is certainly made easier with the use of images and metaphors

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Proofs without dry bones

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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.

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