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Student Reports: A Rewarding Undertaking - Essay Assignments

Frank J. Swetz

          Essay Assignment:  Philosophy   


Mathematicians and scientists have long sought to discover the most unquestionable, impregnable logical foundations on which to base the validity (“truth”) of mathematics. The field of study in which this question is investigated is called the Philosophy of Mathematics or Metamathematics. In the 1930s the work of Kurt Gödel established that no one particular philosophy of mathematics is “better” than any other in the sense that we cannot prove that one philosophical viewpoint on metamathematics is superior to all the others. It is simply a matter of taste which philosophy of mathematics a given mathematician adopts as her or his most fundamental set of beliefs on the subject. There are currently at least 7 Schools of Metamathematics. These primary schools are listed below with a cryptic phrase describing each school.


Pythagorean School  – “All is number.”

Euclidean/Aristotelian School – “The essence of mathematics is to begin with certain declared quotidian and mathematical definitions and premises and then to build up the collection of legitimate theorems in the mathematical system via arguments based upon Aristotelian logic.”

Platonic School – “Mathematical objects already exist and are waiting to be ‘discovered’ in an imperfect ‘shadowy’ fashion by humans.”

Kantian School – “Certain mathematical concepts (e.g., Euclidean geometry) are ‘hard-wired' into the human brain a priori and cannot be circumvented.”

School of Logicism – “To establish a basis for mathematics humans must first develop a ‘sufficiently rich’ system of logic so that mathematics then becomes a branch of that logic.”

School of Formalism – “Mathematics is chiefly concerned with formal symbolic systems which are devoid of content. Mathematics is mostly concerned with questions of the logical ‘consistency,’ ‘independence,’ and ‘completeness’ of its various branches.”

School of Intuitionism (Constructivism) – “Mathematics is to be built solely by finite constructive methods based upon the intuitively given sequence of natural numbers.”


In this essay you are to:


(1) Write a one paragraph description elaborating on the basic tenets of each school listed above concerning its conception of the philosophical foundations of mathematics (which are merely hinted at above by your instructor).

(2) List two or more individuals who were or are strong proponents of the tenets of each Metaphysical School listed above and describe their efforts to promote that school. [In writing this part of the essay, it is certainly OK to use any persons after whom a particular school is named if you so choose.]

(3) Choose the one school (philosophy) of mathematics listed above which you most subscribe to and write a paragraph or two detailing why you prefer that metaphysical viewpoint over the other listed schools’ beliefs.


Frank J. Swetz, "Student Reports: A Rewarding Undertaking - Essay Assignments," Convergence (July 2007)