Truth by Close Proximity

Christian Constanda

At the end of the spring 2020 semester, when teaching was concluded online, I had

an unusual conversation with a student via Microsoft Teams. I will try to reproduce it

as accurately as I can, using the direct dialog style to eliminate redundant verbosity.

Since we, mathematicians, aim to make things as simple as possible, I will abbreviate

myself to P (for professor) and my interlocutor to S (for student). Here is the gist of

the discussion.

S: There has been some controversy lately about mathematics, its nature, origin,

and interpretation, and I would be interested to have your views on the subject.

P: What controversy? I see no controversy where mathematics is concerned. Can

you be more speciﬁc?

S: For example, I read that 2 + 2 should not always be taken to be equal to 4. It

may, perhaps, be equal to 5?

P: It may, if we ban the use of the number 4, counting the positive integers as 1, 2,

3, 5, 6, .... They do that with the thirteenth ﬂoor in some hotels. It’s not as if

there is no thirteenth ﬂoor—they just call it the fourteenth. It’s stupid, and no

educated person should condone this practice.

S: But can we not ﬁnd a different designation for the answer of 2 + 2 without

having to delete the number 4?

P: We certainly can. If we counted in base 4, then we would have 2 + 2 ¼ 10,

pronounced ‘one-zero’ and not ‘ten’.

S: In class, you told us that mathematics is the combination of three fundamental

ingredients...

P: Yes. Numbers, logic, and the power of abstraction. I did say that, and I stand

by it.

C. Constanda (*)

The Charles W. Oliphant Professor of Mathematics, The University of Tulsa, Tulsa, OK, USA

e-mail: [email protected]

A. Wonders (ed.) Math in the Time of Corona, Mathematics Online First Collections,

https://doi.org/10.1007/16618_2020_11

S: You gave me a glimpse of the ﬂexibility of mathematics when it comes to

labelling numbers. Can we not also have some degree of ﬂexibility where logic

is concerned?

P: Logic is deﬁned as a system of principles governing correct and reliable

inference, a particular method of reasoning or argumentation that can be

applied to any branch of knowledge. Flexibility? In mathematics? You mean,

applying a different set of rules than those we have always accepted to prove or

disprove conjectures?

S: Yes, in a certain sense. You just said that logic is a “particular method of

reasoning”. If this ‘particular’ method is replaced by another ‘particular’ one

that is, well, complete and free of self-contradiction, shouldn’t we accept its

conclusions, even though they may appear to differ from what we hold to be

established truth?

P: I cannot say unless I am given the opportunity to examine such a different

system. Do you happen to have one in mind?

S: I do, and it’s very simple to explain.

P: Let’s hear it, then.

S: In the curren t grading scheme, a ﬁnal average of 90 is awarded an A, is it not?

P: It is.

S: But what would you do if someone had an average of 89.5? Would you not be

thinking that perhaps this was due to a possible slight inaccuracy in the

allocation of points to various parts of the solution, or to your mood on the

day as affected by environmental or other subjective factors? Would it not seem

to you a bit inequitable to give an A to someone who scored 90 and a B to

someone who scored 89.5 over six quizzes, three class tests, and a ﬁnal exam?

Would your conscience not feel a pang of guilt if you were to base your

decision on such a minute difference? Would you not be inclined, in the

interest of fairness and justice, to extend the A also to the 89.5 candidate?

P: Depending on the overall circumstances, yes, I might.

S: Very well. We have now established that 89.5 is A-standard. Then, applying

the same logic again, would you not conclude that 89 is also A-standard?

P: I see where you are going. You have just indirectl y enunciated a theorem that I

would call ‘Truth by Close Proximity’, which, for the non-mathematical mind,

could be summarized as “If a is close enough to b, then a is equal to b.”

S: Sounds about right.

P: Unfortunately, there are three essential things that are wrong with your

alternative logic. First, it is based on feelings and emotions, not on cold and

indisputable factual reality. Second, what exactly is meant by ‘close enough’?

What precise, speciﬁc number would quantify this detail? And third—and most

important and dangerous aspect of it—your theorem is open to symmetric

application.

S: I don’

t understand...

P: What is your ﬁnal score in my course?

S: 75.

C. Constanda

P: So I gave you a C, but you are cleverly trying to angle for an A. Let me ask you

this: what grade would an average of 59 attract?

S: An F, I guess.

P: Correct. Using your alternative logic, would you not then agree that 59.5

should also carry an F?

S: But...

P: And then, would not a 60 have the same fate? You see, the same theorem you

used to plead for an A for your 75 average, can be applied to give you an F for

that same score.

S: In that case, perhaps a B...

P: Now, another one o f those factors that go into deciding your grade is that I am

the one who applies the existing rules here. My advice to you is that you should

quit while you are ahead. Take the well-deserved C and concentrate on your

next project. Logic diversity has no place in mathematics. Anyone who thinks

otherwise is either uneducated, or has ulterior motives, not always honorable.

I wonder how this student’ s career will evolve. He is certainly intelligent, has

originality of thought, and is focused on his objective, whether worthy of the effort

or not, which makes me think that some day he might end up in a government

position.

If you would like to read more stories like this, check out my Springer book Dude,

Can You Count?

Christian Constanda

Truth by Close Proximity