In the modern debate about the foundations of mathematics
, Gottlob Frege
and L.E.J. Brouwer
could be considered the scholars most diametrically opposed to each other. Frege, the developer of modern logic
and patriarch of Logicism
, was a staunch realist
when it came to the existence of mathematical objects and truth. They exist objectively and absolutely in the world, and their nature is entirely independent of observer and circumstance. Brouwer, on the other hand, completely rejected the idea of mathematical existence without an observer. He argued for an embodied
, that is wholly based on the system it was derived from—the mind
, the brain
, and conscious
experience. Studying mathematical concepts without somehow implicating the mind would be like studying the notion of property in a world without people.
The two points on which the two men differ the most seem to be the metaphysical status of mathematics in the world and the role of logic in relation to it. According to Brouwer, the mind is the absolute source for all of mathematics. It creates mathematics in time, via a sort of "languageless" operation.1
The very notion of number is a constructed concept that each person must arrive at for himself through reflection
upon temporal experience
. There is no mathematical reality outside of the mind, no truth-values
waiting to be discovered. Moreover, language is simply a tool for communicating the truths that the mind has created. There is no necessary link
, it has no investigatory power, and there is certainly no guarantee that it correctly describes or maps the concepts of mathematics. Because of this, language and hence, logic
, is an impotent descriptive tool that stands apart from mathematical truth. "Proving
" something about math via the syntactic rules of logic would carry about as much weight as one "proving
" that the sky is green simply because the syntax
of English allows for the sentence "the sky is green."
Frege would attack these views on three counts. His first argument would be that by tying everything solely to the brain, Brouwer is making mathematics a contingent
phenomenon. He would say that under this account, people aren’t studying math, they are merely expressing the byproducts of their hormones. What would be keeping math from fluctuating day by day, subject to the whim of our attention
and focus? Math would necessarily be fractured – different versions would exist for each person, differing not only in size but in content. How do we determine which is correct
? How do we know if one person’s concept of 2 is the same as another’s? Frege in fact makes this sort of argument with his famous "hibernation" thought experiment found in the Grundlagen
He envisions a time when all of the sentient
creatures on Earth simultaneously fall asleep and as a consequence, mathematics vanishes. Not only does Brouwer’s conception of mathematics fail to explain its outstanding uniformity and consistency
, it fails to provide the metaphysical tools to create even a coherent
Frege’s second argument would be that there must be something more at work in creating truths than just mental concepts. In the Grundlagen
, he uses the example of our conception of the sun – the small ball we visualize in our head comes nowhere near to approximating the actual reality
of the sun.3
The actual sun is many orders of magnitude greater than our visual image, and there simply is no way that our conception comes even close to approximating the sun’s immense heat. Our mental concept is incorrect on every account, yet somehow we manage to reason correctly about the actual sun. This must mean that we are abstracting
properties from the real entity and then performing truth-preserving operations with this information. Since our mental concept is wrong, truth must be coming from elsewhere. If we were to now apply this reasoning to mathematics, it’s clear that we would run into the same difficulties. Surely one cannot have a clear and accurate conception of a 5000-sided figure. Under Frege’s account, this would not present a problem because his mathematical truths are derived from reasoning
about the properties of the figure. According to Brouwer, however, the figure in the mind is the only truth; so we are left either with no 5000-sided figures or 5000-sided figures whose properties are necessarily not well-defined. Both are unacceptable conclusions.
The last argument that Frege would make on this matter concerns the power of language
. Fundamentally, it is a system of signs
, and words can have meaning solely based on their usage
within this system. It isn’t necessary to form a concept of every meaning, nor does one have to relate the meaning to the world in some way. As long as one understands a word’s usage, one has an understanding of its meaning. Brouwer wants to tie mathematics to the capacity of the mind, but by denying it the use of language, he is denying it one of its most real and useful tools. Without its tools, the mind is necessarily limited in what it can create.
Brouwer would respond to the first argument by saying that creating the concept of number is something that all humans do due to two aspects of their nature. First, humans experience the world through the a priori
perception of time
. This seems to be universally true and appears to be a necessary part of human cognition
. Secondly, everybody is capable of perceiving a period of time as two distinct periods. Whether or not people actually do so and then construct the number system is a matter of their free will, but people seem to do so out of a need to bring order and stability to their world. Brouwer makes it clear that an individual’s realization of mathematics is contingent upon their will and capacity, and that mathematics in theory is a contingent
phenomenon. What happens in practice, however, is that societies (again in the interest of order and stability) train individuals to reason the same way about mathematics, and so a generally coherent system arises that that is mostly consistent amongst individuals.4
In response to Frege’s second argument, Brouwer would most likely say that mental concepts are indeed fuzzy, but the corresponding mathematical objects are definite
because we can produce clear and concise instructions for their construction. The mind has an exact way to build the objects, and that is all that is necessary both for their existence and for their definition
. Brouwer would also respond that he is not opposed to abstraction or reasoning, he is only against reasoning through formal systems
. It is only through formal systems that paradox
can arise – one cannot create a mental concept of an object that simultaneously exists and doesn’t exist, nor can one create a description of how one could construct such an object. Therefore, mental mathematics is sound
, formal mathematics isn’t.
Lastly, Brouwer would respond to Frege’s points on language by chiding the formalists for being too enthusiastic about their systems. Formalists are so quick to posit the efficacy and reality of logic without doing the hard work of really detailing their metaphysics
or determining their specific relation to mathematics. Why are formal systems relevant to mathematics? This is a question, Brouwer says, that the mathematicians leave to the anthropologists
. Which formal system should be chosen as the defining system of mathematical reasoning? This task the mathematicians leave to the psychologists
While the formalists place all of their faith in a typographical
system whose functioning they have no explanation for, Brouwer’s program is providing a framework that is grounded in and derived from a real aspect of the world, human cognition
are all related back to this foundation, while formalism
is based on non-existent principles and blind faith
It is interesting to note that while Brouwer and Frege disagree on the metaphysics
of mathematics, they both agree that it is only intelligible when placed within a context
. For Frege, it is meaningless to look at mathematics outside of the context of linguistic
expressions, and for Brouwer, mathematics is meaningless outside the context of the mind. These views lead directly to their conceptions of the nature of mathematical objects. For Frege, a number is the extension of a linguistic expression and so its existence is in virtue of the expression’s logical form. This means that numbers and mathematical objects are eternal
; they are based on the properties of a relation
and hence are timeless and anybody can manipulate the expression according to the rules of its syntax
, hence it is objective. If it is possible to show through logic that a particular object exists, then it must exist.
Brouwer, on the other hand, holds that mathematical objects are the result of a process
. This means that mathematics is directly related to the specific means of reasoning and is entirely contingent upon their completion. If one doesn’t prove that something exists, then it doesn’t exist, regardless of evidence to the contrary. Because of this, Brouwer’s form of mathematics is more limited in what it considers real – completed infinities
, for example, do not exist because there is no way to construct the entirety of the infinity. Additionally, one is not allowed to use the Law of the Excluded Middle
as a means of proof because while it may "prove" that a number must exist, it never gives a process to arrive/construct the number, which is the ultimate proof in Brouwer's mathematics.
1 Ewald, p. 1180.
2 Benacerraf and Putnam, p. 147-148.
3 Ibid., p. 133.
4 Ewald, p. 1178-1179.
5 Benacerraf and Putnam, p. 81.
Benacerraf, Paul, Putnam, Hilary. Philosophy of Mathematics: Selected Readings Cambridge: Cambridge University Press, 1998.
Ewald, William. From Kant to Hilbert New York: Oxford University Press, Inc. 1999.