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Zachary Balder
June 9, 2014
Logic, Computation, and Understanding:
The Three Roads to an Expanding Reality
William Thurston, in “On Proof and Progress in Mathematics”, and Roger Penrose, in
“Mathematical Intelligence”, address two major questions: how does one arrive at mathematical
truth, and what methods does mathematics include? Thurston and Penrose argue that there are
other methods besides logic and computation that lead to objective truth. Thurston mentions
alternative tools such as language, metaphor, and visualization that one uses to comprehend and
represent mathematical constructs. Penrose concurs that these methods are useful in ascertaining
truth. But are these alternative methods within the scope of mathematics, or are they external to
it? Penrose makes clear that mathematics is a fixed domain that includes forms of non-
computational thought. Thurston, however, paints a different picture; he depicts mathematics as
a network that extends outward to include new methods.
Thurston’s description of mathematics is more convincing than Penrose’s description.
Thurston acknowledges the unpredictable nature of mathematical progress, and he portrays
mathematical practice as a human process. Mathematics is not, as Penrose’s model suggests, a
concrete realm unaffected by human intellectual pursuits. Mathematics bears a strong
resemblance to Thurston’s convoluted system, constantly sprouting in new areas where
mathematicians are exploring.
I disagree with a few claims Thurston and Penrose make. Whereas non-logical methods
are useful for paving new paths of understanding, they must be justified a posteriori with logic.
Thurston is too soft on this point; he argues that if non-logical methods are deductively sound,

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logical validation is unnecessary. Penrose, on the other hand, comes off too strong. Mathematical
understanding has a unique character, he claims, and it transcends computational reduction.
Referencing biology and the environment, Penrose tries to separate understanding from
computation. In the end, his argument is inconsistent. Before elaborating on my criticism, I will
flesh out both arguments, set them next to each other, and see how they interact.
Penrose demonstrates how non-computational methods are useful in understanding
mathematics and reaching objective truth. One non-computational method Penrose highlights is
geometric visualization. Penrose uses a two-dimensional rectangular array, or grid, to verify the
equation, 𝑎 ∗ 𝑏 = 𝑏 ∗ 𝑎 , for all real numbers, a and b (Penrose 122). A rectangular array with
a rows and b columns contains the same number of objects as an array with b rows and a
columns. This number happens to be 𝑎 ∗ 𝑏, or 𝑏 ∗ 𝑎. Counting the number of objects inside
either array is a simple computational procedure. However, the thought of constructing a
rectangular array to prove the equation, 𝑎 ∗ 𝑏 = 𝑏 ∗ 𝑎, is not. According to Penrose, this
process necessitates human intuition and higher-level forms of thought, which transcend mere
calculation (Penrose 115).
Thurston agrees with Penrose’s point that one may use alternative methods to achieve
mathematical understanding. Thurston does not, however, explicitly mention any form of the
word, “computation”. Rather, he discusses logic and its role in mathematics. Thurston and
Penrose cannot speak directly to each other because, when taken literally, they are discussing
different things. For the purposes of this essay, Penrose’s computation and Thurston’s logic can
be viewed through the same lens – as base-level procedural methods, or rules of inference, used
in mathematics. Although Thurston does not explicitly mention computation in his essay, his
discussion of logic has a similar taste to that of Penrose’s.

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Thurston describes several routes, besides logic, that lead to mathematical understanding.
One may think of a derivative as the slope of the tangent line of a graph, as the speed of a
position function of time, as the fraction of infinitesimal change in the value of a function over
the infinitesimal change in the function. One does not have to formulate a logical description of a
derivative to understand one or more of its manifestations. That is, one does not necessarily have
to understand 𝑓′( 𝑥) = lim
h→0
𝑓( 𝑥+ℎ)−𝑓( 𝑥)
ℎ
to understand the derivative as the slope of the tangent
line (Thurston 3). Mathematics is understood in many ways, and this understanding can be
communicated through several different mediums. Informal talks, formal lectures, and academic
papers are all ways through which people convey mathematical understanding (Thurston 6). The
multifarious nature of mathematical understanding, and the multifarious nature of the
communication of mathematical understanding show that mathematics is more than logic.
One may take Thurston’s argument a step further: without using alternative methods, it is
impossible to gain complete mathematical understanding. For instance, one cannot fully
understand the concept of a derivative without understanding the derivative as the slope of the
tangent line or as the speed of the position function. These are separate from logical definitions
(Thurston 3), and non-logical methods for representing a derivative are necessary for holistic
comprehension. In other words, one must look at the picture from every angle to understand it
completely. With only the logical viewpoint, one gains, at best, a partial understanding.
Penrose agrees with Thurston on this point, and he describes a case in which one must
use non-computational methods to reach objective truth. Suppose one has a procedure that
determines whether a calculation, given a certain input, does not terminate. That is, the procedure
terminates if the calculation does not terminate. Using the procedure as the procedure’s input,
one reaches a contradiction. If the procedure terminates, then the procedure does not terminate; if

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the procedure does not terminate, then the procedure terminates (Penrose 130). One cannot use
computation – in this case, the actual procedure – to find the contradiction. Penrose concludes,
“Mathematicians do not simply ascertain mathematical truth by means of knowably sound
calculational procedures” (Penrose 131). Rather, one must use higher intuition to recognize the
impossibility of having a decision procedure for non-terminating calculations.
Penrose and Thurston argue that non-computational and non-logical methods are needed
for reaching mathematical truth and achieving full understanding. Are these methods within the
scope of mathematics? Penrose argues that mathematical thinking, by its nature, contains many
aspects of general intelligence – intuition, common-sense, the appreciation of beauty (Penrose
107). In the first chapter of his book, The Road to Reality: A Complete Guide to the Laws of the
Universe, Penrose describes a platonic world containing all of the mathematical truths (Penrose
11-12). The platonic realm is finite and separate from the mental and physical worlds. To some
degree, the mental world projects onto the platonic realm (Penrose 20). That is, certain human
thought processes lead to objective truth. Penrose characterizes the field of mathematical practice
as a finite world containing those methods that bridge the gap between the mental sphere and the
platonic world. These methods, all of which belong to mathematics, include logic and
computation as well as higher-level thought procedures.
Thurston would agree with Penrose’s point that mathematics includes higher-level
thought procedures. He would, however, disagree with Penrose’s characterization of
mathematics. Thurston defines mathematics as number theory, plane and solid geometry, along
with other subject matter that mathematicians study (Thurston 2). Mathematicians – those
individuals who study mathematics and advance mathematical understanding – are responsible
for setting the mathematical agenda. As human understanding grows more complex,

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mathematics, as a field, expands and annexes the new territory mathematicians are exploring.
Penrose illustrates mathematics as a finite realm containing those methods that link the mental
world to the platonic world. This realm is unaffected by human activity. Thurston, on the other
hand, depicts mathematics as an ever-expanding network driven by human intellectual pursuits
that is always incorporating new methods.
Penrose suggests that mathematics always has, and always will include every method that
leads to objective truth. In the first chapter of The Road to Reality, Penrose claims that Fermat’s
Theorem always has been true, even before a logically-sound proof was formulated (Penrose 14).
That is, Fermat’s Theorem has always existed in the platonic realm, and, assuming there are
humanly-understandable methods for ascertaining Fermat’s Theorem, these methods have
always existed in mathematics. The methods were well-concealed, so it took mathematicians
awhile to uncover them. Nevertheless, they have always been a part of mathematics, according to
Penrose.
Thurston would most likely reject Penrose’s claim that mathematics spans all known and
unknown methods for reaching objective truth. In stark contrast to Penrose’s finite realm of
mathematical methods, Thurston’s illustration of mathematics resembles a tree that is constantly
growing and branching outwards. It is ridiculous to say the sapling is the same as the full grown
tree. It is similarly difficult to say mathematics now is the same as the mathematics of the future.
Mathematicians explore new frontiers and mathematics incorporates new elements. At present,
mathematics cannot possibly contain all potential mathematics because there is no way to predict
in which direction mathematicians will take mathematics. In the distant future, mathematicians
may deem Shakespeare an essential component of mathematical understanding. Thurston’s

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model of mathematics, unlike Penrose’s, accounts for unpredictable advances in mathematical
understanding.
Thurston’s model is a more accurate representation of mathematics than Penrose’s model
because it treats mathematical practice as a human process. Penrose may argue that mathematics
is bound to a certain path, and mathematicians simply follow the trail. This argument seems
unreasonable to me. Penrose might as well say, the sapling is bound to become the tree that it
will become. This is not true because environment is a huge factor. The fertility of the soil and
the availability of water and sunlight play a huge role in the sapling’s growth. If one alters these
factors, the sapling will grow into a different tree or it will not grow at all. Tree growth is
contingent on environmental factors. Likewise, the mathematical agenda is influenced by those
who study mathematics. If the only mathematicians were set theorists, mathematics would look
quite different than if geometric topologists were the only mathematicians. Unlike a computer
program, mathematics is a creative human process that changes as people change.
Mathematicians, ultimately, are the ones who dictate the script. Thurston’s model realizes this
fact while Penrose’s model disregards it.
Though I prefer Thurston’s model of mathematics, I disagree with his view of logical
verification. At one point in the essay, he states, if the toaster works, one does not need to check
the manual. That is, if a proof follows a deductively-coherent argument, one does not need to
check over the formalized structure of the proof (Thurston 7). While non-logical methods are
necessary for ascertaining mathematical concepts, they cannot support the building by
themselves. They provide the basic blueprint for constructing the edifice, though one must revert
to computation and logic – the brick and mortar of mathematics – when verifying the proof.

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While Thurston dismisses the necessity of using logic to evaluate mathematical
understanding, Penrose claims that higher-level thought transcends computational reduction.
Much of human thought is beyond the scope of computation, he argues, and natural selection
favors higher-level understanding (Penrose 134). This view that understanding is biologically
endowed and the human happened to win the evolutionary lottery is overly anthropocentric. I can
accept that human understanding is the product of anatomy and physiology. I cannot accept the
argument that understanding is some special, indefinable human quality perpetuated by natural
selection. Were understanding perpetuated by natural selection, there would have to be some link
to physiological phenomena. While human biological processes are extremely complex, they are
the product of physical law and, to some degree, computation. As follows, understanding would
be a higher-level product of physical law and computation. While it is beyond human capacity to
make sense of the computational reduction of understanding, it exists.
Penrose’s claim that understanding is some intangible quality favored by natural selection
is unreasonable. To make the biological argument, Penrose must link understanding to human
physiology, which is governed by physical law and computation. In doing so, Penrose would
have to concede the existence of a link between understanding and computation. Instead, Penrose
is inconsistent in suggesting that understanding transcends computational reduction.
Penrose then gives an escapist argument – he attributes the un-computational character of
understanding to the un-computational character of the environment (Penrose 135). The
environment is governed by a seemingly infinite number of computations, so randomness seems
to be a reality. But there is always a clear cause and effect. A tree sapling, when placed in
nutrient-rich soil and watered regularly, will grow more healthily than a sapling lacking nutrients
and water. Observing a grove of trees, I may see a random distribution between healthy and sick

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trees. However, when I take all of the minute factors into consideration – the amount of water,
nutrients, and sunlight each tree receives – the randomness does not seem so random. The
appearance of un-computational randomness is due to human inability to perform every
computation and take every factor into consideration. Penrose is wrong to suggest that
understanding transcends computational reduction when the computational reduction exists
beyond the scope of human comprehension.
Penrose and Thurston argue that non-computational and non-logical methods are
necessary for reaching objective truth and gaining understanding. Both authors agree these
methods are part of mathematics. Penrose and Thurston, however, depict mathematics
differently. Penrose illustrates a finite realm containing all methods that lead to objective truth.
This realm is unaffected by human activity, and mathematicians discover rather than create.
Thurston, instead, portrays mathematics as a network that expands outward to include new
methods. Thurston’s model is more accurate because it shows that mathematics proceeds in new,
unpredictable directions. Moreover, Thurston’s model treats mathematical practice as a process
shaped by human intellectual pursuits.
Though I prefer Thurston’s argument, I take issue with claims made by both authors.
Thurston downplays the importance of logically verifying mathematical proofs while Penrose
asserts the irreducibility of higher-level understanding to computational procedures. To
Thurston, I say non-logical methods are useful for constructing new ways of thinking but logic
must be used a posteriori to support them. I say to Penrose, simply because the computational
reduction of understanding is impossible for humans to comprehend or carry out does not mean
it is not there. I do not mean to blur the distinguishing line between computation and higher-level
thought; the two are different even though the latter consists of the former. I think it is beautiful

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that something as complex as understanding is nothing more than levels upon levels of
computations, and yet so much more.