aa r X i v : . [ m a t h . G M ] A p r On Mathematical Ways of Knowing
Gizem KaraaliDept. of Mathematics, Pomona College610 North College Avenue, Claremont, CA 91711, USA [email protected]
Humanistic mathematics is a perspective on mathematics that emphasizes the ways our speciescreates, interacts with, and lives through it. I summarized this idea elsewhere (see [13]) by assertingthat mathematics is the way our species makes sense of this world and that it is inherent in ourthinking machinery; our mathematics in turn is dependent on the way we view our universe andourselves. Lakoff and N´u˜nez [17] argue carefully and eloquently for a mathematics inherently basedon human cognition.Cognition is “the mental action or process of acquiring knowledge and understanding throughthought, experience, and the senses” (see Wikipedia). In this note I attempt to engage with theconstruct of mathematical cognition through the lens of humanistic mathematics. three questions
Cognition is essentially about mental processes involving knowledge, knowing, and understanding;mathematical cognition therefore raises questions about mathematical knowledge, knowing math-ematics, and understanding mathematics. Thus, I first intend to explore broadly three relatedquestions: What does it mean to know something mathematical? How do we come to know a mathematical truth? What does it mean to understand mathematics?In what follows, I will not pretend to offer a comprehensive treatment of any of these questions.But in the very least I intend to open up all three questions in productive ways, so that all readersintrigued by these questions will find in the following assertions worth agreeing with and arguingagainst. question 1
The first question is a natural extension of traditional epistemological investigations into mathe-matics. Philosophers have tinkered with the knowledge question for centuries, or rather, millennia,nd mathematical knowledge has often been a part of the equation. Knowledge as justified truebelief, a core tenet of epistemology since the Enlightenment, is where I want to start this note. If (mathematical) knowledge is justified true belief (in mathematical statements), we have multipleavenues to the first question. Or alternatively we have two related questions to attend to:
What does it mean that a mathematical assertion is true?
What does it mean that a belief in a mathematical assertion is justified?1A is perhaps on the natural playground of mathematicians. Mathematicians seem to be concernedquite single-mindedly and profoundly in the truth of their assertions. One can even suggest thatmathematics is nothing if not true. That is, doing math is making true mathematical assertions. In some sense, therefore, I think that the truth of a mathematical assertion means that it is a partof mathematics, this edifice we human mathematicians are building together. Philosophers havetried to clarify what mathematicians might mean when they say that a mathematical statementis true. Sitting between mathematics and philosophy, the logician Alfred Tarski [24] proposed adefinition of just what truth might mean in “the deductive sciences”, which presumably includemathematics. Once again many have commented on Tarski’s definition of truth. I will not go intothat here but there is indeed much more that can be said along these lines if one is concerned aboutquestion 1A.If we want to address question 1B about justification, we can, like some, invoke idealized conceptionsof mathematical justification involving formal systems and proof theory. But most mathematiciansagree that belief in the truth of a mathematical statement is justified once there is a proof ofthe statement that experts can agree upon. This is quite in tune with Reuben Hersh’s variousdefinitions of a “proof” in [12], most notably “The ‘proof’ is a procedure, an argument, a series ofclaims, that every qualified expert understands and accepts.” Though some philosophers readingHersh might disagree (see for instance [19]), it is indeed the case that when mathematicians claimthat a statement is true, they mean that there is some consensus among the relevant expertsthat the statement is true. And an argument might have been a proof at a given time and placeand afterwards, with contemporary expertise changing sides, it might become invalid. Similarly,proposed proofs do not become proofs until verified and validated by experts. Indeed, one mightargue that “it is the provision of [. . . ] evidence, not the endorsement of experts, that makes [adisplay of symbols, words, diagrams and such] a proof” [19]. However, nothing needs to change inthe display for an argument to remain an alleged proof until experts deem it is valid, and then andonly then is the rest of the mathematical community comfortable in feeling justified to believe thatthe mathematical statement in consideration has finally been proved. What counts as persuasiveevidence is almost always context-dependent. In the case of law this is obvious; even Wikipedia At least since the 1960s with the publication of Edmund Gettier’s “Is Justified True Belief Knowledge?” [7], thisconventional approach to knowledge has seen many rebuttals and rephrasals, but for this note, I will mostly ignorethis recent body of work. My interest is most in line with the idea of mathematical knowledge as being justified truebelief. Truth is surely not the only target of mathematicians. It is not even the driving force, according to WilliamByers, who writes: “Classifying ideas as true or false is just not the best way of thinking about them. Ideas maybe fecund; they may be deep; they may be subtle; they may be trivial. These are the kinds of attributes we shouldascribe to ideas.” [3]. As a working mathematician, I agree with Byers, but this does not mean that I don’t alsobelieve that truth is a prerequisite. Even when mathematicians work with tentative and even patently false assertions,they have a broader truth in mind, and are not done until eventually they can reach that truth. Why do weexpect mathematics to be different?If we see mathematics as something done by humans, the formalist, proof theory-based under-standings of proof and mathematics remain idealized approximations at best. It is the human(or sometimes, and begrudgingly, human-assisted) verification that mathematicians look for in aproof. And this is definitely context-based, both in terms of space-time coordinates and the cul-tural makeup of the audience. As Israel Kleiner quotes in [14], G.F. Simmons wrote “Mathematicalrigor is like clothing: in its style it ought to suit the occasion, and it diminishes comfort and re-stricts freedom of movement if it is either too loose or too tight.” This quote captures well howour understanding of just what should count as proof is dependent on the fashions of our times.This aligns with Harel’s perspective [10]: “Mathematics is a human endeavor, not a predeterminedreality. As such, it is the community of the creators of mathematics who makes decisions aboutthe inclusion of new discoveries in the existing edifice of mathematics.” Among the decisions leftto the human creators of mathematics are the truth of a mathematical statement and the validityof its proof. question 2
The standard modern answer to question 2 is “by a rigorous proof”. Let us leave aside the historicityand cultural dependency of this response now, and its vagueness (what is proof? what is rigor?).I already wrote a bit about all that above. The reader who is not yet convinced may read [14]for more on rigor and proof. But I want to emphasize here the possible distinction between thedoer of the proof and the believer who believes with justification that the proof is valid and thatthe related statement is true. Does the believer need to understand the proof in order to knowthe related statement is true? How similar and how different is this from the calculus studentsaying that they learned calculus because they passed the final exam? Let us narrow things downa bit more. Should the successful calculus student be able to state the Fundamental Theorem ofCalculus? Should they be able to prove it? Should they be able to replicate the argument in theirtextbook or the one their instructor put on the board? Should they be able to provide a convincingargument for its truth? Alternatively, should they be able to use it in a problem that requiresthe result? When do we assume a student has learned, or knows the Fundamental Theorem ofCalculus?Mathematics education researcher Guershon Harel thinks that these kinds of pedagogical questionsare not independent from the philosopher’s concern about mathematical knowledge. In particularhe proposes a definition of mathematics which originates from his pedagogical research that mighthelp us with our endeavor here:Mathematics consists of two complementary subsets: See https://en.wikipedia.org/wiki/Burden_of_proof_(law) for a selection of legal standards of evidence andproof. Do we know if the Four-Color Theorem is true? Yes, we do. Or at least most mathematicians would concedethat the human-assisted computer proof (or alternatively, the computer-assisted human proof) is enough for us toagree that it is true. There are still those who want more human proofs of the result, but the truth of the statementdoes not need further justification. A tangentially related read which might add some insight to this conversation is [23] about the role of rigor andproof in mathematics. The first subset is a collection, or structure, of structures consisting of particularaxioms, definitions, theorems, proofs, problems, and solutions. This subset consists ofall the institutionalized ways of understanding in mathematics throughout history. Itis denoted by WoU. • The second subset consists of all the ways of thinking, which are characteristics of themental acts whose products comprise the first set. It is denoted by WoT. [10].Here Harel uses “ways of understanding” and “ways of thinking” as technical terms. According tohim “a way of understanding is a particular cognitive product of a mental act carried out by anindividual”, while “a way of thinking, on the other hand, is a cognitive characteristic of a mentalact”. Here is how he uses them in context:“Any statement a teacher (or a classmate) utters or puts on the board will be translatedby each individual student into a way of understanding that depends on her or hisexperience and background.”“The range of ways of understanding a fraction makes the area of fractions a powerfulelementary mathematics topic?one that can offer young students a concrete context toconstruct desirable?indeed, crucial?ways of thinking, such as: mathematical conceptscan be understood in different ways, mathematical concepts should be understood in dif-ferent ways, and it is advantageous to change ways of understanding of a mathematicalconcept in the process of solving problems.”Consider the mathematician who reads the statement of the Four-Color Theorem and discussionsabout its proof and as a result is convinced of the veracity of the statement and the sufficiencyof the proof. This mathematician, in my opinion, is not that different from the good studentwho learned of the Fundamental Theorem of Calculus from their instructor, can use it in variousscenarios, can even perhaps outline a convincing argument about why it might be a reasonablething to assume. In each case I’d say the person knows the concept and construct in question.They believe the truth of a true statement and are justified in doing so. They put their trust in arelatively trustworthy source of expertise. But does the student really know (that is, understand)the Fundamental Theorem of Calculus? Does the mathematician really know (that is, understand)the Four-Color Theorem? This naturally brings us to question 3: What does it mean to understandsomething in mathematics? question 3
Harel’s ways of thinking and ways of understanding are related quite visibly to question 3. Forexample, Harel offers a handful of ways of understanding the concept of fractions:a. the part-whole interpretation: m/n (where m and n are positive integers) means “ m out of n objects.”b. m/n means “the sum n + · · · + n , m times”c. “the quantity that results from m units being divided into n equal parts”d. “the measure of a segment m inches long in terms of a ruler whose unit is n inches”4. “the solution to the equation nx = m ”f. “the ratio m : n ; namely, m objects for each n objects.”Similarly, we can develop a list of ways of understanding for the derivative, in terms of the limitdefinition; in terms of slopes of tangent lines; in terms of linear approximations; and so on. And itseems reasonable to assume that when we say that a student understands fractions, we mean thatthey have mastered an indeterminate (but definitely nonzero) number of ways of understanding theconcept. The fluency with which they can move from one interpretation to the other can help usif we want to further qualify how much they understand.Understanding seems to presuppose knowing but is there anything more to it? More specifically,understanding a piece of mathematics does presuppose knowing that piece of mathematics; whatwe want to know is if it involves anything more.This is the perfect context for me to bring up my favorite quote from one of my mathematicalheroes. John von Neumann was a polymath, a genius mathematician who was instrumental inthe development of quantum mechanics, game theory, functional analysis, operator algebras, andcomputer science, a great mathematical mind. This distinguished mathematician is known to havesaid to a young scholar asking for advice: “Young man, in mathematics you do not understandthings. You just get used to them.” Grant that this is a sharp quote. It hits you hard and shakes you up, especially if you have at leasta passing knowledge of the extent of von Neumann’s own mathematical contributions. But can youtake it seriously? Can you get anything out of it in the context of mathematical knowledge andmathematical ways of knowing?My personal take on this quote is twofold.One is that of the optimistic student of mathematics. Even if I feel like I am not understandingsomething, there is some benefit to pushing forward, if only a bit more. Doubtless the greatmathematician is right; sometimes you simply have to move on, after accepting the fact as a factand see where it leads you. Stubborn patience. Dogged perseverance.However perhaps von Neumann did not really mean to recommend moving forward without under-standing. Perhaps he was saying something else, that what you call understanding is not somethingsubtle or sublime. In fact, when we are learning a new concept, a new theory, don’t we start bymaking mental patterns, charting new pathways in our mind, formatting our minds so certain typesof programs run well or smoothly enough? How is this different cognitively from getting used tobrushing our teeth before going to bed, splashing our faces before leaving the bathroom, eatingwith the fork in our right hand? Habit forming is done by doing something over and over again;aren’t mathematical ways of thinking and ways of understanding reinforced by repeated practiceas well? And is there a genuinely different, a genuinely distinct sense of “understanding” that goesbeyond “getting used to thinking of the concept in question in a particularly productive manner”?Imagine a student who learns to think of a complex number first as an ordered pair, then as apoint on the complex plane, and then as a linear transformation on the complex plane. Whencan we say the student knows complex numbers? When do we say the student understands them?I agree with Emily Grosholz who argues, using complex numbers as a concrete case study, that“the best way to teach students mathematics is through a repertoire of modes of representation, For an interesting discussion of this quote see https://math.stackexchange.com/questions/11267/what-are-some-interpretations-of-von-neumanns-quote . mathematical ways of knowing In the remainder of this note, I want to delineate a construct I will call “mathematical ways ofknowing”. This is in some sense related to what Harel calls “ways of thinking”. But I believe it isnot exactly the same.I start by the axiomatic definition that mathematics is one of the main systematic bodies of knowl-edge that formulates and occasionally aims to address questions about human perception of theworld and the human endeavors to understand it. The concepts and constructs of number, shape,form, time, change, and chance are fundamental to our understanding of our world as well as our-selves. These concepts and constructs are mathematical in nature, or at least they are naturallyamenable to mathematical approaches.Within this setting, I mean by a mathematical way of knowing a way of formulating and address-ing a question or a set of questions about our world and ourselves that allows for mathematicalinquiry. It might be interesting to try and put these mathematical ways of knowing in contrastto or in conversation with a handful of other ways of knowing: scientific / empirical, faith-based,philosophical. I do no such thing in this note however. Here I merely put down some ideas asplaceholders, as tentative yet suggestive notes to a self that might or might not be able to comeback to revisit them in a possible future. The formatting of the brain might sound strange but is not that far from the point of education. The point ofeducation is to shape students’ minds. What is that if not brain formatting? Indoctrination might also fit in thisview. Some might take this to discussions of how mathematics education promotes dogmatic beliefs. Though I aminterested in such inquiry, I will not pursue it here. athematical ways of knowing: rationalism and imagination The first two mathematical ways of knowing I would like to consider are rationalism and imagina-tion.Rationalism in mathematics is the fundamental assumption that we ought to reach our mathe-matical truths through reason. Experiment and experience might provide hints towards a truth,but they are never enough to convince us in the final count. Even if we “know something in ourguts”, we are not convinced we have mathematical certainty until we can reason our way to thatsomething.And reason and rational thinking are captured effectively in axiomatic thinking. The central tenetsof axiomatic thinking are captured by Tarski in the following:When we set out to construct a given discipline, we distinguish, first of all, a certain smallgroup of expressions of this discipline that seem to us to be immediately understandable;the expressions in this group we call primitive terms or undefined terms, and we employthem without explaining their meanings. At the same time we adopt the principle:not to employ any of the other expressions of the discipline under consideration, unlessits meaning has first been determined with the help of primitive terms and of suchexpressions of the discipline whose meanings have been explained previously. [25]Thus, we begin with some initial assumptions, ideas, fundamental beliefs, core values, axioms. Wetake certain things for granted. We try to make these as self-evident as possible. And from therewe build our argument step by step, using logic as our guide. We define new constructs in terms ofolder, already accepted ones, and thus attempt to build a new world which has a solid foundation.Mathematical rationality can be found in the various versions of the ontological argument for theexistence of God, as well as in the Declaration of Independence of the United States of America(see [8] for a convincing argument about how mathematical rationality is built into the Declarationas well as many other illustrative examples of the impact of mathematical ways of knowing onWestern thought).In the history of intellectual thought Rationalism of the European Enlightenment was met witha backlash movement, Romanticism. Today we can but do not have to see these two as directlyopposing and mutually exclusive methodologies, each rejecting and invalidating the other. Alter-natively, and I believe more productively, we might choose to accept that they point to two distinctways of knowing, and occasionally certain truths will be more accessible via one way than another.Mathematical rationalism also has a similar complement, in what I will call mathematical imagina-tion, or mathematical romanticism, if you will. Mathematical imagination is the way we select ouraxioms, the way we fix our principles. Mathematical imagination is how we determine our targettruths. Human mathematicians do not start with a random formal axiomatic system and automat-ically go through all possible provable truths of the theory determined by it. Instead they engagewith the worlds around them, both real and imaginary, and detect what is interesting, conjecturewhat might be productive to pursue, and then set out. Our human mathematics has a freedom toit and mathematical imagination captures this freedom.And freedom, broadly construed, may be viewed in these terms as well. S´andor Szathm´ari’s utopian,satiric novel,
The Voyage to Kazohinia , might just be the best (fictional) guide to how mathematics But of course, self-evidence, just like beauty, is in the eye of the beholder. There is more that can be said here.
7s fundamental to a human society. Susan Siggelakis [21] describes how the protagonist of the novellearns that in a society without mathematics, “nothing stable exists with which a human canconnect and find meaning in his/her life”. There is only chaos and violence. As Edward Frenkelsays in [6] “where there is no mathematics, there is no freedom”. mathematical ways of knowing: universals and eclecticisms
Two other mathematical ways of knowing beckon us here: Universals and eclecticisms. The ten-dency of the mathematician to generalize is well known. “Mathematics compares the most di-verse phenomena and discovers the secret analogies that unite them,” wrote Jean Baptiste JosephFourier. “The art of doing mathematics is forgetting about the superfluous information,” saysHendrik Lenstra. Thus the human mathematician tries to generalize, to abstract from specificexamples, to reach universal statements with “any” and “all” that capture the essence of what istrue about a whole slew of eclectic examples. The human desire to “see the big picture”, the humantendency to “find patterns” is precisely what I mean by mathematical universalism.Accompanying and complementing (again productively) this tendency is the alternative, what I willcall mathematical eclecticism. David Hilbert describes the complementarity of these two tendenciesas follows:In mathematics, as in any scientific research, we find two tendencies present. On theone hand, the tendency toward abstraction seeks to crystallize the logical relationsinherent in the maze of material that is being studied, and to correlate the material ina systematic and orderly manner. On the other hand, the tendency toward intuitiveunderstanding fosters a more immediate grasp of the objects one studies, a live rapportwith them, so to speak, which stresses the concrete meaning of their relations.Thus by mathematical eclecticism I mean the search for that one representative example on theone hand (“the art of doing mathematics consists in finding that special case which contains allthe germs of generality,” wrote Hilbert), and the excitement of the weirdness of eclectic cases onthe other. In fact a lot of mathematics concerns itself with concrete examples. Paul Halmos wrote,“the heart of mathematics consists of concrete examples and concrete problems.” John B. Conwaywrote, “mathematics is a collection of examples; a theorem is a statement about a collection ofexamples and the purpose of proving theorems is to classify and explain the examples. . .”“We think in generalities, but we live in details,” said Alfred North Whitehead. In fact, I believe wethink in both and we live in both. Once again, these two mathematical ways of knowing complementone another and help us live our human lives. mathematical ways of knowing: certainty and ambiguity
Alan Lightman says it best [18]: “We are idealists and we are realists. We are dreamers and we arebuilders. We are experiencers and we are experimenters. We long for certainties, yet we ourselvesare full of the ambiguities of the Mona Lisa and the I Ching. We ourselves are a part of the yin-yangof the world.” Certainty and ambiguity find their way into mathematics and mathematical ways ofknowing in a similarly complementary fashion.Certainty is a part of most people’s perception of mathematics. Mathematics, for those people, ismade up of questions and answers. Answers are certain once we know them. Indeed mathematics8s perhaps the only certain knowledge we will ever have. I do not wish to minimize this perspective.I admit that I too have a romantic attachment to the idea that mathematical knowledge has aquality of certainty that goes farther than any other type of knowledge. And it took us a long timeto get over this perspective as the unique way to conceive of mathematical truths and mathematicalknowledge.The loss of certainty in mathematics began more than a century ago [15]. Lakatos was also influ-ential in convincing many who cared to listen of the fallibility of mathematics [16]. And Byers [3]is perhaps the most detailed expositor of the role of ambiguity in the work of mathematics today.The complementarity of these two ways of knowing is rich and, at least to me, inspiring. applications of mathematical ways of knowing
Doing math at school or anywhere else is tied deeply in with our views of ourselves. This has goodand bad aspects of course. We can relate our mathematical experiences to confidence, resilience,and determination as well as feelings of inadequacy, resistance, and rebelliousness; and as manycan personally attest, any combination of the six can occur together. There is much emotionin mathematical engagement: hatred, love, anger, fear, anxiety, surprise, frustration, anticipation.How we handle mathematical challenges (suffering alone, valiantly standing defeated or undefeated,finding commiserators and conspirators) tells us about ourselves. Many of those who continue to domath after school connect with it at an emotional and personal level. We find aesthetic stimulationand creative joy in mathematical activity, as well as terrible frustration and occasional bouts oftedium.But can mathematical ways of knowing allow us to reach self-knowledge and a sense of identitymathematically? Andres Sanchez recounts how through an intentional application of set theoryto his own personal life, he was able to discover his true identity and sexuality [20]. Set theory,more generally, offers us pathways of thinking about belonging and not belonging. A clever studentof mine, when asked to form study groups, chose to name his team “the identity element” as hewanted to work alone for the project in question. Mathematical ways of knowing may come inhandy when thinking in terms of borders and boundaries of nations, communities, and culturalgroups. They can help us think of inequalities and intersectionalities in different ways [4]. Theycan help us with moral dilemmas [27, 28]. Mathematical ways of thinking and knowing can indeedallow us to view and understand ourselves in new and insightful ways. parting words – till next time. . . We have come a long way since Ptolemy argued that “mathematics alone yields knowledge andthat, furthermore, it is the only path to the good life” [5]. Mathematical knowledge has lost itscertainty somewhere along the way, and primality in the eye of the public a long time ago. Math-ematicians eventually learned to be humbler about the reaches of mathematical ways of knowing.But mathematics can still yield powerful knowledge, and not just the kind that can blow up citiesand optimize factory production. I urge us to try and open up to the world once again. If we digdeeper into mathematical ways of knowing and the contexts where they might apply, mathematicsmay yet surprise us. And in turn other ways of knowing can help us understand our mathematics better [11]. eferences [1] Anderson, Lorin W.; Krathwohl, David R., eds. (2001). A taxonomy for learning, teaching,and assessing: A revision of Bloom’s taxonomy of educational objectives. Allyn and Bacon.[2] Bloom, B. S.; Engelhart, M. D.; Furst, E. J.; Hill, W. H.; Krathwohl, D. R. (1956). Taxonomy ofeducational objectives: The classification of educational goals. Handbook I: Cognitive domain.New York: David McKay Company.[3] Byers, W. (2007). How Mathematicians Think: using ambiguity, contradiction, and paradox tocreate mathematics . Princeton: Princeton University Press.[4] Eugenia Cheng, TEDx talk on how abstract mathe-matics can help us understand the world, available at ,last accessed on April 15, 2019.[5] Feke, Jacqueline (2018). Ptolemy’s Philosophy: Mathematics as a Way of Life. PrincetonUniversity Press.[6] Frenkel, Edward (2013). Love and Math: The Heart of Hidden Reality. Basic Books.[7] Edmund Gettier, “Is Justified True Belief Knowledge?”,
Analysis , Volume Issue 6 (1963),pages 121–123.[8] Grabiner, Judith V. (1988). ”The Centrality of Mathematics in the History of WesternThought,” Mathematics Magazine, Volume 61 Issue 4, 220-230.[9] Grosholz, Emily (2013). “Teaching the Complex Numbers: What History and Philosophy ofMathematics Suggest,” Journal of Humanistic Mathematics, Volume 3 Issue 1, 62-73.[10] Harel, Guershon (2008). “What is Mathematics? A Pedagogical Answer to a PhilosophicalQuestion”. In Bonnie Gold and Roger Simons (eds.), Proof and Other Dilemmas: Mathematicsand Philosophy. Mathematical Association of America. 265-290.[11] Guti´errez, Rochelle (2012). “Embracing Nepantla: Rethinking ‘Knowledge’ and its Use inMathematics Teaching”, REDIMAT - Journal of Research in Mathematics Education, Volume1 Issue 1, 29-56.[12] Hersh, Reuben (2014). Experiencing Mathematics: What Do We Do, When We Do Mathe-matics? Providence: American Mathematical Society.[13] Karaali, Gizem (2015). “Can Zombies Do Math?” In Mariana Bockarova, Marcel Danesi,Dragana Martinovic and Rafael Nunez (eds.), Mind in Mathematics: Essays on MathematicalCognition and Mathematical Method, Interdisciplinary Studies on the Nature of Mathematics
Proofs and Refutations: the logic of mathematical discovery . New York:Cambridge University Press.[17] Lakoff, G. and N´u˜nez, R. E. (2000).
Where Mathematics Comes From: How the EmbodiedMind Brings Mathematics into Being . New York: Basic Books.[18] Lightman, Alan (2018). Searching for Stars on an Island in Maine. Penguin Random House.[19] Pollard, Stephen (2014). “Review of Experiencing Mathematics by Reuben Hersh”, PhilosophiaMathematica, Volume 22 Issue 2, 271-274.[20] Sanchez, Andres (2018). ”My Sets and Sexuality,” Journal of Humanistic Mathematics, Volume8 Issue 1, 359-370. https://scholarship.claremont.edu/jhm/vol8/iss1/18/[21] Siggelakis, Susan J. (2019). “Sndor Szathmri’s Kazohinia: Mathematics and thePlatonic Idea”, Journal of Humanistic Mathematics, Volume 9 Issue 1, 3-23.https://scholarship.claremont.edu/jhm/vol9/iss1/3/[22] Smith, M.S., & Stein, M.K. (1998). “Selecting and creating mathematical tasks: From researchto practice.” Mathematics teaching in the middle school, Volume 3 Issue 5, 344-350[23] Terence Tao, “There?s more to mathematics than rigour and proofs”, blog pot available at https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ ,last accessed on April 15, 2019.[24] Tarski, Alfred (1933/1956). “The concept of truth in the languages of the deductive sciences”(Polish), Prace Towarzystwa Naukowego Warszawskiego, Wydzial III Nauk Matematyczno-Fizycznych 34, Warsaw; reprinted in Zygmunt 1995, 13-172; expanded English translation inTarski 1983 [1956], 152-278.[25] Tarski, Alfred (1946). Introduction to Logic and the Methodology of the Deductive Sciences.Oxford University Press.[26] Tarski, A. and Vaught, R., 1956, “Arithmetical extensions of relational systems”, CompositioMathematica, Volume 13, 81-102.[27] Sarah Voss, “A Workshop to Introduce Concepts of Moral Math”,
Journal of Hu-manistic Mathematics , Volume Issue 2 (July 2012), pages 114–128. Available at: https://scholarship.claremont.edu/jhm/vol2/iss2/10 , last accessed on April 15, 2019.[28] Sarah Voss, “Fuzzy Logic in Health Care Settings: Moral Math for Value-Laden Choices”,
Journal of Humanistic Mathematics , Volume Issue 2 (July 2016), pages 161–178. Available at: https://scholarship.claremont.edu/jhm/vol6/iss2/12https://scholarship.claremont.edu/jhm/vol6/iss2/12