TOK RESOURCE.ORG - 2024

AREAS OF KNOWLEDGE:
MATHEMATICS:
PLATONISTS AND FORMALISTS

Just a minute WARM-UP

The activities on this page build on the content in This statement is false, which, to a significant extent, sabotaged the notion of absolute certainty and a solid foundation to all of mathematics.

CLASS ACTIVITY I: IS MATH INVENTED OR DISCOVERED?

Do not reveal the writing assignment texts for the second class activity in advance.
Arrange students in random triplets and have them sit together to ensure eye contact. Ask them to nominate a spokesperson who will later report back their findings to the entire class. Echoing the warm-up, and their configuration, announce that they now have a timed three minutes to reach consensus in addressing the following Knowledge Question:

• Is mathematics in invented or discovered?

If convenient, use a timer with an loud, dissonant alarm. When the three minutes are up, terminate the group conversations abruptly. Next a spokesperson from each triplet will report back. Allow questions and comments but keep it brisk.

The following additional generative questions may or may not be necessary to facilitate engaging, heartfelt discussion:

• What just happened?

• Was the strict timing unfair and counterproductive?

• Was it unreasonable to expect consensus?

• Is there a single right or wrong answer to this question?

If time permits, and the idea has not already emerged spontaneously, try:

• Would an advanced alien intelligence recognize our math?

CLASS ACTIVITY II: WRITING ASSIGNMENT

Begin with a silent reading of the following texts (from Davis, Philip J. and Hersh, Reuben (1981) The Mathematical Experience. Houghton Mifflin, Boston and New York.) Next solicit four student volunteers to perform public readings. Some highlights in bold font have been added for emphasis.

TEXT #1: PLATONISM (p.318)

According to Platonism, mathematical objects are real. Their existence is an objective fact quite independent of our knowledge of them… They exist outside the space and time of physical existence. They are immutable—they were not created, and they will not change or disappear.  Any meaningful question about a mathematical object has a definite answer, whether we are able to determine it or not. According to Platonism, a mathematician is an empirical scientist like a geologist; he cannot invent anything, because it is all there already. All he can do is discover.   

TEXT #2: FORMALISM (p. 339-40)

Mathematics from arithmetic on up is just a game of logical deduction. The formalist defines mathematics as the science of rigorous proof… either we have a proof or we have nothing.  Any logical proof must have a starting point. So, a mathematician must start with some undefined terms. These are called “assumptions” or “axioms”… So far as pure mathematics is concerned, the interpretations we give to the  axioms is irrelevant. We are concerned only with valid logical deductions from them... they are free of any possible doubt or error, because the process of rigorous proof and deduction leaves no gaps or loopholes.  

TEXT #3: WHAT REAL MATHEMATICIANS DO (p. 354)  

The actual situation is this… we have real mathematics, with proofs which are established by “consensus of the qualified”… Even to the “qualified reader,” there are normally differences of opinion as to whether a real proof (i.e., one that is actually spoken or written down) is complete or correct. These doubts are resolved by communication and explanation…

If a theorem is widely known and used, its proof frequently studied, if alternative proofs are invented, if it has known applications and generalizations and is analogous to known results in related areas, then it comes to be “regarded as rock bottom.” In this way of course, all of arithmetic and Euclidean geometry are rock bottom.

TEXT #4: HUMANIST MATH (p. 410-11)

Mathematics is not the study of an ideal, preexisting non-temporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather it is the part of human studies which is capable of achieving science-like consensus, capable of establishing reproducible results…

Mathematics does have a subject matter, and its statements are meaningful.  The meaning, however, is to be found in the shared understanding of human beings, not in an external nonhuman reality. In this respect, mathematics is similar to an ideology,  a religion, or an art form; it deals with human meanings, and is intelligible only within the context of culture. In other words, mathematics is a humanistic study. It is one of the humanities.

The special feature of mathematics that distinguishes it from other humanities is its science-like quality. Its conclusions are compelling, like the conclusions of natural science. They are not simply products of opinion, and not subject to permanent disagreement like the ideas of literary criticism.

As mathematicians we know that we invent ideal objects, and then try to discover the facts about them. Any philosophy which cannot accommodate this knowledge is too small… This means accepting the legitimacy of mathematics as it is: fallible correctible, and meaningful

WRITTEN ASSIGNMENT QUESTION

With reference to the Davis and Hersh texts, to what extent do Platonism and Formalism provide a rock solid foundation for mathematics?

Limit your response to 800 words.