Open Response to the USPTO — Physical Aspects of Mathematics
by PolR, author of An Explanation of Computation Theory for Lawyers.
USPTO has issued a request
for comments on their new interim guidance, Interim Guidance for Determining Subject Matter Eligibility for Process Claims in View of Bilski v. Kappos [PDF]. They invited comment by Monday, September 27, from the public on three questions in particular:
1. What are examples of claims that do not meet the machine-or-transformation test but nevertheless remain patent-eligible because they do not recite an abstract idea?
They ask in effect how to tell an abstract idea from an
application of the idea. This article suggests answers to that question from the perspective of a computer professional.
2. What are examples of claims that meet the machine-or-transformation test but nevertheless are not patent-eligible because they recite an abstract idea?
3. The decision in Bilski suggested that it might be possible to "defin[e] a narrower category or class of patent applications that claim to instruct how business should be conducted," such that the category itself would be unpatentable as "an attempt to patent abstract ideas." Bilski slip op. at 12. Do any such "categories" exist? If so, how does the category itself represent an "attempt to patent abstract ideas?"
The Supreme Court has reaffirmed
the trilogy of cases Gottschalk v. Benson, Parker v. Flook and Diamond v. Diehr
concerning the non-patentability of mathematical algorithms:
In Benson, the Court considered whether a patent application for an algorithm to convert binary-coded decimal numerals into pure binary code was a "process" under §101. 409 U. S., at 6467. The Court first explained that "'[a] principle, in the abstract, is a fundamental truth; an original cause; a motive; these cannot be patented, as no one can claim in either of them an exclusive right.'" Id., at 67 (quoting Le Roy, 14 How., at 175). The Court then held the application at issue was not a "process," but an unpatentable abstract idea. "It is conceded that one may not patent an idea. But in practical effect that would be the result if the formula for converting . . . numerals to pure binary numerals were patented in this case." 409 U. S., at 71. A contrary holding "would wholly pre-empt the mathematical formula and in practical effect would be a patent on the algorithm itself." Id., at 72.
In Benson and Flook the unpatentable algorithm is
software. But it is recognized that an algorithm may take the
form of hardware or software.
In another case, In Re Alappat, the Federal Circuit had
acknowledged the possibility that a patent on a circuit could
read on a computer programmed with software and that this
could make the algorithm unpatentable. They eventually ruled
the circuit was patentable on the basis that they thought
(among other things) the algorithm was not a mathematical
algorithm in the sense of Benson, Flook and
Diehr but this doesn't change the fact that they first
considered the possibility of going the other way. (Also note the dissenting opinions in Alappat, which raised precisely the danger of patenting mathematical discoveries.)
In Flook, the Court considered the next logical step after
Benson. The applicant there attempted to patent a procedure for monitoring the conditions during the catalytic conversion process in the petrochemical and oil-refining industries. The application's only innovation was reliance on a mathematical algorithm. 437 U. S., at 585586. Flook held the invention was not a patentable "process." The Court conceded the invention at issue, unlike the algorithm in Benson, had been limited so that it could still be freely used outside the petrochemical and oil-refining industries. 437 U. S., at 589590. Nevertheless, Flook rejected "[t]he notion that post-solution activity, no matter how conventional or obvious in itself, can transform an unpatentable principle into a patentable process." Id., at 590. The Court concluded that the process at issue there was "unpatentable under §101, not because it contain[ed] a mathematical algorithm as one component, but because once that algorithm [wa]s assumed to be within the prior art, the application, considered as a whole, contain[ed] no patentable invention." Id., at 594. As the Court later explained, Flook stands for the proposition that the prohibition against patenting abstract ideas "cannot be circumvented by attempting to limit the use of the formula to a particular technological environment" or adding "insignificant postsolution activity." Diehr, 450 U. S., at 191192.
Finally, in Diehr, the Court established a limitation on the principles articulated in Benson and Flook. The application in Diehr claimed a previously unknown method for "molding raw, uncured synthetic rubber into cured precision products," using a mathematical formula to complete some of its several steps by way of a computer. 450 U. S., at 177. Diehr explained that while an abstract idea, law of nature, or mathematical formula could not be patented, "an application of a law of nature or mathematical formula to a known structure or process may well be deserving of patent protection." Id., at 187. Diehr emphasized
the need to consider the invention as a whole, rather than "dissect[ing] the claims into old and new elements and then . . . ignor[ing] the presence of the old elements in the analysis." Id., at 188. Finally, the Court concluded that because the claim was not "an attempt to patent a mathematical formula, but rather [was] an industrial process for the molding of rubber products," it fell within §101's patentable subject matter. Id., at 192-193.
The court in Diehr held an industrial process to cure
rubber that uses an algorithm is patentable. This court made a
distinction between the algorithm and a non-algorithmic process
that comprises the algorithm as one of its elements.
This article discusses a specific aspect of this issue.
Could a patent on a circuit, as in Alappat, or a patent on
physical computational activity, such as in Benson and
Flook, be considered patents on an abstract idea? This is a
notion that has a sound basis in mathematics. The purpose of this
article is to explain this basis and show some possible problems
that occur when one does not take it into consideration.
Physical Tools Are Necessary to the Practice of
relationship between mathematics and the physical world is often
described, outside of the Supreme Court trilogy, with two notions:
together these ideas suggest that whenever we have a relationship
between mathematics and the physical world, mathematics is always used
to provide an abstract disembodied model describing the reality. This
view, however, is not always accurate.
a pocket calculator. If one calculates the energy contained in matter
with the formula E=mc2, does the formula
describe the calculator? Of course not. The relationship goes the
other way round. The calculator is a tool to do mathematics, not an
application of the formula. I like to say the calculator is a
physical model of the abstract mathematical calculation. Instead of
doing the calculation in our head, we enter the numbers in the machine
and watch what it does. The calculator shows us how the mathematics
idea that mathematics can model the physical world is sometimes used
to discount the use of mathematics in a patent claim. The argument is
that it is not because an invention can be described mathematically
that it is abstract and non-patentable, because otherwise everything
is mathematics and nothing would be patentable. This is a fair point,
but what if the modeling relationship goes the other way round? What
if, like a pocket calculator, the physical entity is used to represent
the mathematical abstraction? Isn't this the very issue raised by the
Supreme Court trilogy? It is hard to
see how the questions raised by Benson,
Flook and Diehr
can be answered without first getting an understanding of which
direction the modeling relationship goes and what are the
So to summarize, when there is a relationship between mathematics and a physical entity this relationship may occur in two different ways:
1. Mathematics is used to provide a model of the physical reality. This is what happens, for example, when we use mathematics to describe the laws of physics.
The task of sorting out whether or not a patent claim covers an algorithm requires that we understand which of these two alternatives is the correct one. These two situations are what I call the directions of the modeling relationship. In the first situation the direction is from the abstract to the concrete. In the second situation the direction is from the concrete to the abstract.
2. A physical device of process is used to carry out an abstract mathematical operation. This is what happens, for example, when we use a calculator to carry out a calculation.
notion that the mathematics of computing are related to the computer
in the same way as the laws of physics are related to physical
phenomenon is erroneous1.
In computation theory, the modeling relationship goes
the other way round. This follows from common sense by observing a
pocket calculator and the use of pencil and paper. This is also
supported by established knowledge in abstract mathematics, computer
science, and philosophy.
Another way to make the same point is to establish that the use of a physical symbolic representation is an essential requirement of the practice of mathematics with references to mathematical literature. Without some physical representation no mathematical activity can occur at all. If one is not careful, it is possible that a patent on a physical device happens to be a patent on mathematics. This is the outcome that Benson, Flook and Diehr are meant to avoid.
The simplest explanation of this point is that mathematics is a written discipline. One cannot use mathematical concepts without
writing them down except in the most simple situations. The written
text is necessarily physical. But mathematicians don't care whether
the text is written on paper or is electronic signals such as bits.
As long as there is a usable physical representation mathematical
activity can occur.
this simple explanation doesn't do justice to the actual mathematical
truth. This is not just about the ability of text to express meaning.
It is also about the ability to conceive. One cannot think of or use
a mathematical idea in the first place if it cannot be written.
There is a considerable body of mathematical research on the topic
of the connection between what can be written and which mathematical
truths can be understood by humans. There are some areas of
mathematics that are inaccessible because the written language is
insufficient even though its use is unavoidable.
I provide later in this article a selection of the main points with
reference to mathematical literature in case lawyers would like
authoritative evidence. But it is not the details of the
mathematical research that matter, although some of them are
legally relevant for purposes other than the main topic of this
article. It is that the research has actually been made. We have
factual evidence that one cannot think of or use a mathematical
idea in the first place if it cannot be represented physically,
and this is what gives
Benson, Flook and Diehr a sound mathematical
Application to Patent Law: Analysis of In Re
software is a recent technology it doesn't bring a new type of
relationship between the abstract and the concrete. On the contrary,
the same sort of relationship has existed since antiquity. The
is an ancient
mechanical computer from the first century BC. The Romans used
abacuses. More modern devices are the slide rulers and the
analyzers. All these devices are physical models for abstract
mathematical computational procedures.
Although the range of
applications and the economic consequences of software are arguably
the relationship of the physical world and the abstract mathematics
my knowledge, this is a fact that hasn't been acknowledged by the
courts. This has led to erroneous analysis of the subject matter
claimed in patents. A good example is the In re Alappat case.
Its examination in the light of a correct understanding is
instructive. The litigated patent is a simple one, and the
mathematical aspects are clear cut. It is also a historically
important case because it introduced the useful, concrete and
tangible result test. This test was found to be unworkable for
empirical reasons. After years of usage, it was found that this test
allows patents that shouldn't be.
When we are
equipped with a correct understanding of the relationship between
mathematical abstraction and the physical circuit, it is possible to
see *why* the useful, concrete and tangible result test doesn't lead to
a correct analysis of a patent.
Alappat claim covers a rasterizer for an
The oscilloscope is a device for measuring electromagnetic signals
and showing the waveforms visually on a screen. With modern digital
electronics, the picture is drawn pixel by pixel on a rectangular
grid. Without the rasterizer, the curved lines of the waveform have an
awkward visual appearance because the pixels are drawn either
entirely lit or entirely black and must be located at the
intersections of rows and columns on the grid. These restrictions
give the curve a discontinuous or jagged appearance. The rasterizer
makes the curve appear smooth by averaging the luminosity of
neighboring pixels. This makes the waveform more visually pleasing to
patented claim is directed to a digital electronic circuit. The
Alappat court quotes the claim as follow:
circuit carries out a computation. The symbols are digital 0s and 1s.
They are used to represent numbers that have real-world meanings,
specifically the sample magnitudes of an input waveform, the computed
anti-aliased pixel intensity data, and other numbers that are required
in the course of the computation. The circuitry performs the
computation according to mathematically defined rules. The rules are
not stated in the patent claim although they can presumably be found
in the patent specification3.
Examining the components reveals their function is to carry out
ordinary arithmetic. The two arithmetic logic circuits perform
absolute value functions to determine distance of numbers. Barrel
shifters multiply (or divide) by powers of two and the ROM stores a
lookup table containing values of illumination intensity data that
are precomputed according to a mathematical function. There is
nothing in the circuit that is not doing an arithmetical calculation.
However this arithmetic is thinly veiled by the mention of the
meanings of the numbers in the claimed functions.
independent claim 15 is construed in accordance with Section 112
Para. 6, claim 15 reads as follows, the subject matter in brackets
representing the structure which Alappat discloses in his
specification as corresponding to the respective means language
recited in the claims:
rasterizer [a “machine”] for converting vector list data
representing sample magnitudes of an input waveform into anti-aliased
pixel illumination intensity data to be displayed on a display means
[an arithmetic logic circuit configured to perform an absolute
value function, or an equivalent thereof] for determining the
vertical distance between the endpoints of each of the vectors in the
[an arithmetic logic circuit configured to perform an absolute
value function, or an equivalent thereof] for determining the
elevation of a row of pixels that is spanned by the vector;
[a pair of barrel shifters, or equivalents thereof] for
normalizing the vertical distance and elevation; and
[a read only memory (ROM) containing illumination intensity
data, or an equivalent thereof] for outputting illumination intensity
data as a predetermined function of the normalized vertical distance
is evident, claim 15 unquestionably recites a machine, or apparatus,
made up of a combination of known electronic circuitry elements.
issue in front of the court was whether this claim was patentable
subject matter because it is a circuit or whether it is non-patentable because it would pre-empt some mathematics as per Benson,
Flook and Diehr.
view as a computer professional is that we can answer
the issue in front of the court just by looking at the circuit. It
does arithmetic calculations and produces numeric answers. Arithmetic
and numbers are clearly mathematics.
court didn't look at the circuit the way I would. They looked at the
text of the claim. Upon analyzing the claim the court concluded the
rasterizer is not doing a mathematical calculation with this
reasoning. (emphasis in the original):
the foregoing, the proper inquiry in dealing with the so called
mathematical subject matter exception to Section 101 alleged herein
is to see whether the claimed subject matter as a whole is a
disembodied mathematical concept, whether categorized as a
mathematical formula, mathematical equation, mathematical algorithm,
or the like, which in essence represents nothing more than a “law
of nature,” “natural phenomenon,” or “abstract idea.” If
so, Diehr precludes the patenting of that subject matter. That
is not the case here.
notice the use of the expression “disembodied mathematical
concept”. This is a phrase that is inconsistent with the notion
that an abstract mathematical calculation must be carried out
physically in order to be used at all. Then the court concludes the
circuit is not an abstract idea because it yields a useful, concrete
and tangible result.
many, or arguably even all, of the means elements recited in claim 15
represent circuitry elements that perform mathematical calculations,
which is essentially true of all digital electrical circuits, the
claimed invention as a whole is directed to a combination of
interrelated elements which combine to form a machine for converting
discrete waveform data samples into anti-aliased pixel illumination
intensity data to be displayed on a display means. This is not a
disembodied mathematical concept which may be characterized as an
“abstract idea,” but rather a specific machine to produce a
useful, concrete, and tangible result.
is the useful, concrete and tangible result of the circuit? Again my
answer as a computer professional is to look at the circuit. It carries out an
arithmetic calculation and produces numeric answers in the form of
bits. This is the result. They are the electronic equivalent of the
symbols a mathematician writes on paper. The bits are electronic
signals that are useful, concrete and tangible. And still they are
numbers produced by an arithmetic calculation. When I look at the
circuit I don't see how the useful, concrete and tangible result test
separates mathematics from something that is not mathematics, because
the symbols produced by any computation will always be useful,
concrete and tangible.
court didn't follow my approach. They didn't look at
the circuit. They looked at the text of the patent claim. The court
found that the display of the bits on a display is relevant. They
also find that the meaning of the numbers as quoted in the patent
text are relevant. The above analysis contains this text:
claimed invention as a whole is directed to a combination of
interrelated elements which combine to form a machine for converting
discrete waveform data samples into anti-aliased pixel illumination
intensity data to be displayed on a display means.
is a very different from my look-at-the-circuit approach. Does the
display really matter? When I look at the circuit I don't find any
display. According to the claim, the rasterizer is comprised of four
elements listed from (a) to (d), and the display is not among them. By
itself the rasterizer has no capability of displaying anti-aliased
pixel illumination intensity data on a display. For this you need an
apparatus such as the oscilloscope where the rasterizer is connected
to a display. But this claim, as I read it, is claiming just the rasterizer and not
the entire oscilloscope. Do I misread something?4
course one may connect a rasterizer to a display as the claim
preamble suggests. This is only an option because nothing in the laws
and principles of electronics requires it. The rasterizer may as well
be connected to a hard disk to record the result of the calculations
if someone wishes. Or some other creative use could be found. The
useful, concrete and tangible result contemplated by the court is one
that may occur but will not necessarily occur. Its occurrence depends
on how exactly the rasterizer is used.
Perhaps I am overly focused on the display. Perhaps they just point out that the bits being produced are not any random bits but they are bits representing pixel illumination intensity data. Perhaps the meaning of the bits somehow makes the bits a useful, concrete and tangible result in the eyes of the court. But then the correct understanding of the modeling relationships doesn't support the court conclusion. There are two modeling relationships as follows:
1. The rasterizer circuit is a physical model for the abstract calculation. This is a relationship from the concrete to the abstract.
These are two distinct modeling relationships going in opposite directions and their concrete elements are different. This court is using one modeling relationship to reach a conclusion about the other. But the two relationships are independent because the circuit doesn't have to be connected to a display and the mathematical calculation doesn't have to be about pixel illumination data. The court's logic doesn't follow from how the modeling works.
2. The calculation is an abstract mathematical model (formula) for the illumination of pixels on a screen. This is a relationship from the abstract to the concrete.
court further explains (bold added):
fact that the four claimed means elements function to transform one
set of data to another through what may be viewed as a series of
mathematical calculations does not alone justify a holding that the
claim as a whole is directed to nonstatutory subject matter. See
In re Iwahashi,
888 F.2d at 1375, 12 USPQ2d at 1911. Indeed,
claim 15 as written is not “so abstract and sweeping” that it
would “wholly pre-empt” the use of any apparatus employing the
combination of mathematical calculations recited therein.
409 U.S. at 68-72 (1972). Rather,
claim 15 is limited to the use of a particularly claimed combination
of elements performing the particularly claimed combination of
calculations to transform, i.e., rasterize, digitized waveforms
(data) into anti-aliased, pixel illumination data to produce a smooth
court worked from the idea that the real-world meaning of the bits
defines the actual computation. We can immediately see how this causes
a problem. Please refer to the sentence in bold. How does this work?
If an electronic engineer wants to build an apparatus employing the
combination of mathematical calculations recited therein, he has to
build the same circuit as the one described in the Alappat
claim. Or in the alternative he needs some other circuit that has the
same functionality and will be subject to the treatment patent law
applies to functional equivalents. This is the very situation Benson
is meant to avoid.
the claim preamble's recitation that the subject matter for which
Alappat seeks patent protection is a rasterizer for creating a smooth
waveform is not a mere field-of-use label having no significance.
Indeed, the preamble specifically recites that the claimed rasterizer
converts waveform data into output illumination data for a display,
and the means elements recited in the body of the claim make
reference not only to the inputted waveform data recited in the
preamble but also to the output illumination data also recited in the
preamble. Claim 15 thus defines a combination of elements
constituting a machine for producing an anti-aliased waveform.
is an example of how this problem would happen. Suppose our
hypothetical engineer designs a new oscilloscope whose display draws
the waveforms in yellow over a neutral gray background5.
There is no change in pixel illumination intensity value between the
background and the waveform. It is a change of color. The
anti-aliasing is done by altering the color saturation value to make
the yellow color more or less grayish. A fully saturated color is
pure yellow and a totally unsaturated color is pure gray. If we use
the same formula for the anti-aliasing, the rasterizer no longer
computes pixel illumination intensity data. It computes pixel color
saturation data. But the same circuit is being used.
is a change from the overall function of the rasterizer, because the
preamble says “converting vector list data representing sample
magnitudes of an input waveform into anti-aliased pixel illumination
intensity data”. This doesn't happen any more. This is also a change
from element (d) which will turn out not to be present. This element
says “means for outputting illumination intensity data”. This
will not occur. The new rasterizer will output color saturation
arithmetic formula used to perform the anti-aliasing is the same. But
the meaning of the bits is changed. In order to carry out this
calculation, the same circuit is used. The question is, does it
infringe? If the answer is yes, then how is it that “claim 15 as
written is not 'so abstract and sweeping' that it would 'wholly
pre-empt' the use of any apparatus employing the combination of
mathematical calculations recited therein”? Conversely, if the circuit
doesn't infringe, what exactly is being covered by the patent? Isn't
it supposed to claim a circuit?
the language about waveform data samples, anti-aliased pixel
illumination intensity data and the likes imposes no limitations to
the circuit beyond constraining the structure to represent the
correct arithmetic calculation according to the intended
interpretation of the numbers7.
The fact that the numbers mean pixel illumination intensity data is
not recorded with the bits. If the same calculations are required for
another purpose the bits will be the same, the arithmetic operations
will be the same, and the circuit carrying out the task will be the
same. Therefore anyone who needs this calculation has no choice but
to use the same circuit (or an equivalent).
reason is this: the calculation is done with mathematical symbols.
When I write 24, nobody can tell just by looking at the written digits
whether this is the number 24 in the abstract, a count of 24 apples,
or 24 monkeys. This is contextual information that is not recorded
with the written digits. When someone adds numbers, the pocket
calculator doesn't care if the numbers mean dollars in a bank account
or something else. It just adds the numbers. The real life meaning of
the numbers is not stored with the symbols, and it is not used by the
same thing happens with bits. They are the electronic version of
mathematical symbols. Whether they represent pixel illumination
intensity data or or the quantity of jet fuel required by a plane at
take off is not recorded with the bits. The bits just represent
numbers, and the digital electronics components work just the same
whatever the meaning is. If an engineer needs to carry out the same
computation for another purpose, the circuit that will fulfill this
purpose is the same circuit8.
course when an engineer uses the rasterizer circuit for another
purpose, the circuit is no longer a combination of elements
constituting a machine for producing an anti-aliased waveform as the
court puts it. In this sense, it is no longer the machine claimed by
Alappat claim 15. This doesn't change the fact that the same
circuit is used.
think that in addition to claiming an abstract mathematical
computation, this claim is somehow faulty. I am not a lawyer. This is
a hunch and not knowledge. My hunch is that Section
112 paragraph 2 might be applicable. (emphasis added):
specification shall conclude with one or more claims particularly
pointing out and distinctly claiming the subject matter which
the applicant regards as his invention.
hunch is that reliance on the real-world meaning does not distinctly
claim the subject matter unless the real-world meaning is actually
used by an element of the claim and not merely referred to. If I
understand the law correctly, patent claims that do not conform to
section 112 paragraph 2 are invalid and cannot be enforced.
A Note on the Machine or Transformation Test
the questions asked, the USPTO requests examples of patents that
pass the machine or transformation test but are patents on abstract
ideas. We have just seen such an example.
rasterizer passes the machine or transformation test, because it is a
patent on a specific circuit. But it is a patent on a mathematical
computation as surely as the Benson patent is. Such a remark
is applicable every time a specific circuit is built for the purpose
of making a computation pursuant to a mathematical algorithm and
there is no invention outside of the computation.
machine or transformation test should work fine in most if not all
circumstances where the abstractions involved are models for a real-world invention, like the application of the laws of physics to
engineering. Then this test will provide a strong clue as to whether
the patent is on the abstract model or on the real-world application.
But this test doesn't work when the modeling relationship is reversed,
because in such case the machine or process is the tool used to
understand the abstract ideas. Then it is possible that a patent on
the tool is effectively patenting the idea as was the case in Benson
is very difficult to make sense of the notion that a patent on a
circuit is a patent on an abstract idea when we assume the modeling
relationship always works from the abstract to the physical and never
the other way round. The trilogy Benson, Flook and
Diehr is best understood after the modeling relationship has
been worked out properly.
Appendix: Mathematical References
Here are the references to published literature that establish
that mathematics require a physical representation of the concept
in order to be able to understand, use, or even conceive mathematical
ideas. Physical tools impose fundamental limitations on the ability
of human beings to reach mathematical truths. My hope is that lawyers can use these
references to locate authorities to support this point in court.
very first two paragraphs of the very first chapter9
of Introduction to Metamathematics10
Cole Kleene are (emphasis in the original)
turning to our main subject, it will be appropriate to notice briefly
Cantor's theory of sets.
first thing Kleene does is to use a concrete and tangible procedure
to represent the abstract concept of cardinal number in Cantor's
theory of sets. What is a cardinal number? On page 9 Kleene quotes
flock of four sheep and a grove of four trees are related to each
other in a way in which neither is related to a pile of three stones
or a grove of seven trees. Although the words for numbers have been
used to state this truism on the printed page, the relationship to
which we refer underlies the concept of cardinal number. Without
counting the sheep or trees, one can pair them with each other, for
example by tethering the sheep to the trees, so that each sheep and
each tree belongs to exactly one of the pairs. Such a pairing between
the members of two collections or 'sets' of objects is called a
one-to-one (1-1) correspondence.
describes them thus: “The general concept which with the aid of our
active intelligence results from a set M, when we abstract
from the nature of its various elements and from the order of their
being given, we call the 'power' or 'cardinal number' of M.”
is definitely an abstract concept. Cantor's theory of sets is a
mathematical theory. Although this concept of cardinal number is
described using English there is a more formal definition given
mathematically by means of an operation of pairing which is called
numerically challenged shepherd may use pairing to keep track of his
sheep. The shepherd may be unable to count past three, but he knows
that if a tree ends up without a sheep, then a sheep is missing. It
doesn't matter if the pairing is done “in the abstract” in the
shepherd's mind or if it is done in the concrete by physically
tethering the sheep to trees. Both ways of pairing count sheep.
this little example the abstract theory of sets is not used as a
model of the act of tethering sheep to trees. It goes the other way
round. The act of tethering is a physical model for the mathematical
theory of cardinal numbers. It allows the shepherd of the example to
overcome his limited mind and perform an abstract operation he would
otherwise be unable to do.
people are able to count normally. All of us are subject to ordinary
human limitations. There are abstractions that cannot be conceived or
handled without physical assistance. This is why we use pocket
calculators or pencil and paper for all but the simplest
ancient Greeks practiced mathematics, especially geometry. They used
figures drawn in the sand or on papyrus to help them track the
geometrical abstractions. As Plato
puts it in book 7 of The Republic (around 380 BC):
also know how they make use of visible figures and discourse about
them, though what they really have in mind is the originals of which
these figures are images: they are not reasoning, for instance, about
this particular square and diagonal which they have drawn, but about
the Square and the Diagonal; and so in all cases. The
diagrams they draw and the models they make are actual things, which
may have their shadows or images in water; but now they serve in
their turn as images, while the student is seeking to behold those
realities which only thought can apprehend.
we use the written text to achieve the same result. Like the drawn
figures, written text is a physical tool.
is more at play than the ability of text to express meaning. The
tools also enable procedures, the algorithms, for solving problems.
The ancient Greeks carried out their mathematical algorithms mostly
with compass and straightedge. Modern mathematicians use either
pencil and paper or a computer. For example we all learned in school
the algorithms to carry out additions, subtractions, multiplications
and divisions by manipulations of decimal digits written with pencil
and paper. Algebra, calculus and other mathematical disciplines
provide methods for solving problems based on the written text. These
methods can be automated by means of digital electronics devices like
the pocket calculator and the computer.
choice of the physical tool is important. This choice limits which
abstractions we are able to use and determines how easy the
computational process will be. If you have any doubt please try to
multiply MDCCVIII by CCXXXIV with pencil and paper using roman
numerals for all manipulations. Please don't cheat. No translation
into decimal numbers is allowed.
is another example. Try solving x2 + 3x - 2 = 0 like you
were taught in school but using only English words instead of the
algebraic notations. You need to use sentences like “the square of
the value added three times the value from which you subtract two
yield nothing.” According to Kleene12,
this is how mathematical problems were solved before the invention of
the algebraic notation by Vieta and others. The discovery of new
notations is one of the ways mathematics makes progress. Our ability
to use algorithms that solve the problem at hand is contingent on
the availability of the adequate physical tool.
issue goes deeper than mere convenience and practicality. Some
problems are fundamentally unsolvable with some categories of tools
regardless of whether we may resolve the practical aspects. A famous
example is the squaring
of the circle problem in Euclidean geometry. As Howard Delong
when discussing the power of modern algebraic methods when applied to
geometry: (emphasis in the original)
this powerful method of analysis, nineteenth century scientists were
finally able to solve the three famous outstanding geometric problems
which the Greeks passed on to posterity: the duplication of the cube,
the squaring of the circle, and the trisection of an arbitrary angle.
The problem was to make an exact construction with the aid of a
compass and straightedge alone. It was shown in each case that it is
impossible to solve the problem under the condition stated.
Impossible here means logically impossible. That is,
given certain assumptions, it was shown that it is as impossible to
solve these problems as it is to draw a square circle. The proofs are
complex — especially the one for the squaring of the circle — but
they nevertheless settle the problems once and for all.
is another example. There are real numbers that can't be defined by
any mathematical formulas. The reason is that formulas are countably
infinite while real numbers are uncountably infinite. This is
a limitation of the algebraic notation as physical symbols written on
infinite means an enumeration like the natural numbers 0, 1, 2, 3 …
If an infinite set can be enumerated in such a way each member can be
associated with a natural number, then it is countably infinite14.
Formulas can be put in lexicographical orders, this is something like
a, b, … z, aa, ab, ac, … zz, aaa, aab etc. except that the order
must be defined on the mathematical symbols rather than the Latin
alphabet. You group formulas by their length first, and then you sort
the formulas of the same length in the lexicographical order. The
result is an enumeration that could be numbered15.
Formulas are countably infinite.
is proven that real numbers are not countably infinite. The proof is
given by the Cantor diagonal method16.
Assuming you have an ordering of real numbers you prove there is a
real number not of the list as follows.
write the infinite decimal expansion of each real number in the
enumeration as in the above example. If some real number doesn't have
an infinite expansion then you pad it with infinitely many zeros
(like the third number in the example). Then you take the first digit
of the first number, the second digit of the second number and you
continue like this over the diagonal. You make a new number with the
rule: every time the diagonal digit is 5, you make the new number a
6, otherwise you make it a 5. With the example this gives:
result is a real number guaranteed not to be in the list. In other
words, there can be no enumeration of all the real numbers, because
given any enumeration, we can always find at least one real number
that is not included.
consequence of the Cantor diagonal method is that there are real
numbers we cannot define and comprehend because there is no formula
able to define them. This is a limitation of relying on a
mathematical language made of written symbols.
tools do more than representing meaning and computing solutions to
problems. They are also the embodiment of the rules of logic. This is
a topic that has been studied by philosophers and mathematicians
since the nineteenth century. The name of the discipline is symbolic
logic. Its most prominent features are propositional calculus and
predicate calculus. They are synthetic languages made of written
symbols whose purpose is to provide a logical foundation to
mathematics. The formulas and expressions in these languages must
follow a rigidly defined syntax, like a programming language. This
formalism allows to subject the language itself to mathematical
study. This is called metamathematics, or mathematics about
of the greatest mathematical discoveries of the twentieth century
arose from metamathematics. For instance see this series of
first incompleteness theorem: in every formal system of
mathematics that is powerful enough to represent the arithmetic of
natural numbers there is a formula that is true but cannot be
theorem: consider a formal system that represents pure ordinary
mathematical logic (this is known as predicate calculus),
there is no algorithm that will decide whether or not a formula in
this system can be proven as a theorem.
theorem: if a formal system is powerful enough to represent the
arithmetic of natural numbers then it simultaneously supports an
alternative interpretation where there are “unnatural numbers”,
that is number other than those in the familiar sequence 0, 1, 2,
3… to infinity.
have translated into plain English the mathematical terms of arts and
notations in order to make their meaning accessible to laymen. Please
go by the mathematical text for any use that requires mathematical
theorems and others like them are called limitative theorems. They
establish fundamental epistemological limits to what can be known by
man. This is a fascinating topic of deep philosophical consequences.
This is an area of knowledge where mathematics, philosophy and
computer science overlap. Delong, philosopher and author of A
Profile of Mathematical Logic, describes these epistemological
limits as follow19:
(emphasis in the original)
the limitative theorems represent then is the discovery of an
abstract structure which is of such a sort that it is impossible for
any human to make systematically complete and correct assumptions
about it. No matter how hard we try, we cannot talk about just the
natural numbers, but must always talk about the unnatural numbers,
too. In other words, just as our sensory perceptions have limits
which can be extended but not eliminated by certain technique of
science (for example, the microscope), so our abstract conceptions
are limited and the methods of mathematics which are intended to
extend them (for example, mathematical induction) are at best partial
in their effect. Our powers of conceptual discrimination are no less
limited than our powers of perceptual discrimination.
Delong proceeds to examine what happens when we question this
assumption. The topic is quite fascinating, but I don't want to stray
too far away from the legal issues. This quote will be enough for our
doesn't seem to be any way around these conclusions. But it should be
noted that the above argument depends on what is known as realism
(or platonism) in mathematics. That is, it is assumed that the
abstract structure of arithmetic exists independently of human
conceptions about it. The assumption is embedded in classical
mathematics, but we might question it.
is not so much the theorems that are relevant but the way they have
been derived. Howard Delong explains20:
(emphasis in the original)
can be argued that: “I object,] the impossibility of squaring a
circle or trisecting an angle is not philosophically interesting. It
is perfectly possible to do it, but not with just a straightedge and
a compass. Similarly, when Gödel proves that there are undecidable
formulas, or Church that there is no decision procedure, or Skolem
that a calculus doesn't categorically represent the natural numbers,
all this comes to is that using the means they have selected (just
as Euclid selected a straightedge and compass), their conclusions
follow. There is no philosophical import to this: it just suggests
that mathematicians and logicians must look for other means.”
point is that fundamental limits on the extent of mathematical truths
that can be known to mankind follow from the physical tools that are
necessary to the practice of mathematics.
objection would have a great deal of sting were it not for one
circumstance: There do not appear to be any other means.
To see this, let's consider briefly the requirements we make in order
to claim deductive knowledge. We first require that all the “symbols”
which are used be publicly and effectively recognizable. The word
“symbols” is in double quotes because it is not required that
they be symbols in the usual sense of the word, that is, marks on
paper or on a blackboard, etc. They might be sounds or colors or
electric charges. The requirement includes not only being able to
tell the difference between a symbol and something else (for example
we should be able to tell the difference between a parenthesis and a
broken fishhook), but being able to effectively recognize repeated
occurrences of the same symbol type [for example, '(' is of the same
symbol type as '(']. The 'all' in our requirement means that the
number of symbols be finite (for example, a zillion symbols) or, if
infinite, that there be an ordering (called the alphabetic order
above) such that only a finite number of symbols precede any given
symbol in that ordering.
we require that the length of any given sentence be finite and that a
sentence be effectively recognizable as such. We make the same
requirement of rules of inference. As for the axioms, we require that
they be either finite or, if infinite, that there be an ordering such
that only a finite number of axioms precede any given axiom in that
order. For the theorems, we do not require that there be an effective
method to recognize them, but only that there be a constructive
program which, if carried out, will produce any given theorem.
Finally, we require that for any given sentence we be able to
effectively recognize a proof of it if one should be presented. The
above are all syntactical requirements. There is only one semantical
requirement: that the theorems all be true under every interpretation
which makes the axioms true.
gives a mathematically exact definition to each element identified as
a requirement of mathematics by Delong. Then mathematical methods are
applied to these definitions to prove the limitative theorems and
other results. This is an inquiry on the very nature of the tools
required to gain deductive knowledge, essentially the symbols and the
various syntactical elements that are made with them. This inquiry
goes well beyond the basic observations that the written text has a
meaning, like is done in the printed matter legal doctrine. It
extends the analysis to computational and deductive processes which
exist in the real world as manipulation of symbols and syntactic
is the sort of thing the law should look at in the context of
Benson, Flook, and Diehr. It is not possible to
fully understand the relationship of mathematical abstractions and
physical reality without knowing what is found in this part of
mathematics. It not only relates to why a patent on a circuit may
be a patent on an abstract idea, it also relates to why software
is speech because it involves the study of the relationship between
text, computation, and meaning. Its very topic is the ability of
mankind to reach knowledge by means of written symbols.
the features of metamathematics there are objective definitions of
what is a mathematical proof21.
These definitions define 'proof' by syntactic means. It is the
organizing of the formulas as syntactic elements in the proper
sequence that makes a mathematical proof valid. Semantics, in the
sense of looking at the meaning of the formulas and verifying their
truths, plays no part in these definitions. A proof is correct when
it satisfies syntactic rules. The consequence is that mathematical
reasoning is amenable to machine processing. Kleene observes22:
is known as automatic
theorem proving. This is the automation of operations of abstract
logic. Quoting from the linked page: (emphasis and bold in the
may be applied in metamathematics to seeking proofs of theorems, or
to checking proposed proofs etc.
Theorem Proving (ATP) deals with the development of computer
programs that show that some statement (the conjecture) is a
logical consequence of a set of statements (the axioms
and hypotheses). ATP systems are used in a wide variety of
domains. For examples, a mathematician might prove the conjecture
that groups of order two are commutative, from the axioms of group
theory; a management consultant might formulate axioms that describe
how organizations grow and interact, and from those axioms prove that
organizational death rates decrease with age; a hardware developer
might validate the design of a circuit by proving a conjecture that
describes a circuit's performance, given axioms that describe the
circuit itself; or a frustrated teenager might formulate the jumbled
faces of a Rubik's cube as a conjecture and prove, from axioms that
describe legal changes to the cube's configuration, that the cube can
be rearranged to the solution state. All of these are tasks that can
be performed by an ATP system, given an appropriate formulation of
the problem as axioms, hypotheses, and a conjecture.
is an example of using the real-world device to model abstract
logical processes. The linked page discusses more, pointing to actual
programs and applications.
feature of metamathematics is the precise definition of what is
computable. This is what Benson, Flook, and Diehr
refer to as a “mathematical algorithm”. This is knowledge which
should be used by the law. This definition follows from the same type
of epistemological inquiry as the limitative theorems. Delong
(emphasis in the original, the occurrence of patent statute section
numbers is pure coincidence):
begin by imagining some human who is faced with a specific
computational problem. It might be some such problem as computing the
sum of 101 and 102 and 103 and . . . and 198 and 199, where the three
dots indicate one occurrence of all the natural numbers between 103
and 198. We assume that he is working according to a finite set of
rule which have been fixed before the problem was given and that he
is using pencil and paper. We also assume that after a finite amount
of time he stops with the correct answer.
too we see the prominent position of the use of symbols in
mathematical processes. The analysis of how a human computer can be
replaced by an automatic device has been carried out by Alan Turing.
The outcome is the Turing machine24.
we examine this paradigm case of computing with a view toward
eliminating inessential features, a number of such features comes to
mind, such as the use of pencil and paper or the particular
computational problem chosen. However the most striking one appears
to be the human: the computer might as well be a machine.
von Neumann was familiar with Turing's work and used it to derive the
stored-program architecture which forms the basis of nearly all
modern programmable computers25.
Jack Copeland reports on The
Turing Archives for the History of Computing:
1944, John von Neumann joined the ENIAC group. He had become
'intrigued' (Goldstine's word) with Turing's universal machine while
Turing was at Princeton University during 1936-1938. At the Moore
School, von Neumann emphasised the importance of the stored-program
concept for electronic computing, including the possibility of
allowing the machine to modify its own program in useful ways while
running (for example, in order to control loops and branching).
Turing's paper of 1936 ('On Computable Numbers, with an Application
to the Entscheidungsproblem') was required reading for members of von
Neumann's post-war computer project at the Institute for Advanced
Study, Princeton University (Julian Bigelow in personal communication
with William Aspray, reported in the latter's John von Neumann and
the Origins of Modern Computing Cambridge, Mass.: MIT Press
(1990), pp. 178, 313).
computers relate to written text in a way analog technologies
implementing computations don't. Digital information is made of
symbols, the bits, the 0s and the 1s which represent the boolean
values False and True respectively. By design, the same class of
computational processes that could, in principle, be carried out by
means of pencil and paper are automated by the computer. This is in
accordance to the analysis of computation done by Alan Turing and
the engineering decision made by John von Neumann and his colleagues.
is more than enough evidence that physical devices and computations
have the ability to represent abstract ideas. Their use is essential
for human beings to even be able to conceive, let alone use, these
ideas. In particular the computations described by algorithms must be
carried out physically in order to be used at all. If you patent the
physical means to carry out the computation then you may end up
patenting the computation itself if you are not careful. This is the
mathematical basis for the non-patentability of algorithms as per
Benson, Flook and Diehr.
v. Kappos [PDF] [text] The Supreme Court opinion
450 U.S. 175, 182 (1981)
U.S. 63, 71-72 (1972)
U.S. Court of Appeals Federal Circuit, 33 F.3d 1526 July 29, 1994
re Bilski the Court of Appeals for the Federal Circuit decision
703 F.2d 1381
437 U.S. 584 (1978)
2000] Davis, Martin, Engines of Logic, Mathematicians and the
Origin of the Computer, W.W. Norton and Company, 2000. This book
was originally published under the title The Universal Computer:
The Road from Leibnitz to Turing.
1970] Delong, Howard. A Profile of Mathematical Logic.
Addison-Wesley Publishing Company. 1970. Reprints of this book are
available from Dover Publications.
Kurt, 1986, Collected
vol. 1, Oxford: Oxford University Press.
1952] Kleene, Stephen Cole, Introduction to Metamathematics,
D. Van Nostrand Company, 1952. I use the 2009 reprint by Ishi Press
1967] Kleene, Stephen Cole, Mathematical Logic, John Wiley &
Sons, Inc. New York, 1967. I use the 2002 reprint from Dover
1936] Turing, Alan, On Computable Number with and Application to
the Entscheidungsproblem, Proceeding of the London Mathematical
Society, ser. 2, vol 42 (1936), pp. 230-67. Correction: vol 43 (1937)
paper could be ordered from the publisher here [Link]
paper is available on-line here [link]
are from the MacTutor
History of Mathematics Archive
Knuth's 1994 letter to the US Commissioner of Patents and Trademark:
“An algorithm is an abstract concept unrelated to physical laws
of the universe.”
shouldn't underestimate the importance of the written text for the
storage, communication and processing of information. Before the
advent of computers written text was as important as computers are
nowadays. It still is. The social, scientific and economic issues
raised by software are not nearly as new as students of software
patents may think. Computers represent an expansion of existing
capabilities and not the creation of new capabilities that didn't
exist before. This expansion is formidable, I will agree, but an
expansion of existing capabilities no matter how formidable is not
the same thing as the creation of new capabilities.
text of the court ruling contains a description of the mathematical
converting vector list data representing sample magnitudes of an
input waveform into anti-aliased pixel illumination intensity data
to be displayed on a display means” implies that the act of
displaying occurs after the rasterizer has produced its output. It
appears to me that the display is explicitly excluded.
or not the engineer has an awful taste in matters of color is beside
design may have an amusing twist. In this particular setup the blue
component of the video signal happens to have a well defined
mathematical relationship with color saturation. When the blue
component has a high value it indicates the pixel corresponds to the
gray background because pure yellow doesn't have any blue component.
And conversely a zero value of the blue component indicates pure
yellow. What if the engineer modifies the display to monitor the
blue component and adjust the pixel illumination intensity inversely
proportional to the blue component? Low blue value raises the
intensity to the brightest yellow while high blue values drop the
intensity to pitch black. Then we display the waveform as a yellow
line on a black background just as the Alappat rasterizer
might do. The color saturation rasterizer would control pixel
illumination intensity without computing pixel illumination
intensity data. I am bringing up this point just for its amusement
value. This scenario is beside the point of this article. Lawyers
may wish to discuss whether this modified display makes a functional
equivalent of the Alappat circuit and whether this will lead
the point needs further hammering, such calculations can be done by
a standard off-the-shelf pocket calculator when only the human being
operating the calculator knows the meaning of the numbers.
the patent been on the process manipulating the bits then by the
same logic assigning a different meaning doesn't make a physical
process distinct from the base mathematical computation.
[Kleene 1952] p. 3.
1952] in the reference section.
[Kleene 1952] p. 9 where the exact mathematical definition is given.
1971] p. 69.
 pp. 3-6.
1967] p. 178. This is assuming there is a finite number of allowable
symbols in a formula. If the number of allowable symbols is infinite
(not a practical possibility) then the formulas are still countable
but the ordering requires the use of Gödel numbers. See [Gödel
 pp. 6-8, [Kleene 1967] pp. 180-183 or [Delong 1971] pp.
the the topic of [Kleene 1952] which is titled Introduction to
Metamathematics. Chapter IV to X are especially applicable. This
is also covered in [Kleene 1967] chapters I to III and in [Delong
1971] chapters 3 and 4.
1971] p. 195.
1971] p. 204
1971] pp. 193-194.
is known as proof theory. See for example [Kleene 1967] pp. 33-58,
107-134. There are several proof theories depending on the system of
logic being used. Each of them is mathematically unambiguous.
1967] p. 201.
1971] p. 197.
analysis of the use of symbols by humans is found in section 9 of
1936]. His thesis is that all computations done with pencil and
paper can be replicated by Turing machines, therefore Turing
machines are a mathematically accurate description of what is
history is told in [Davis 2000] chapters 7 and 8.