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Logic for Leadership: Why Executives Should Study Math (Part 2)

If you’re just picking up here, you might want to go back and read Part 1. In this three-part series, we are exploring the merits and misconceptions of mathematics to help us understand why studying math can give an edge to people outside of the field, particularly, those in business.

In this section, we will explore what math and logic have to offer.

What does math have to offer?

Let’s pick up at the example we used in the last section. Recall the poor math student, who mistakenly asserts that 2 + 2 = 5 and is brushed off by the teacher, who should have instead encouraged her to explore why 5 could not be arrived at within the confines of our logical system. Recall also the still confused, but much better off, chemistry student who thinks that HO2 is water and receives a comprehensive explanation from her teacher.

When taught in the common way, the science student receives a better explanation than the math student, coming away with a stronger understanding not only of the correct answer but of the system in which the question functions. The science student is taught why we arrive at that answer, for, while the class may not conduct an experiment to show the molecular composition of water, she receives an explanation of how one might arrive at such an answer.

Yet, if taught in the right way, the mathematics student is able to better explore the boundaries of the system on her own, not relying directly on what others (teachers, scholars) tell her about the truths of that system. The beauty of math is that it is broadly accessible; all you need is a pencil, paper, and perhaps a calculator or protractor. By teaching math in an exploratory light, students are able to peruse the boundaries of thought and logic with or without a guide, forming neurological pathways around pattern recognition and sematic connections in a distilled form.

Symbolic Logic

One digestible derivative of mathematics, which may allow students to more easily explore its applications and boundaries, is symbol logic. If you aren’t familiar with it, think of logic as a translation of a verbal argument into a symbolic system that reveals syntactical strings and deductive steps (or lack thereof). Does your argument hang together? Have you outlined your premises and followed logical rules to reach your conclusions?

The below is an example of a basic logical argument, using the correct symbols.

1. Premise: A OR B → C — It is a rule that if either I write this article (A) or if my colleague writes this article (B) then this article will be published (C).
2. Premise: A exists — We take as a given that I will write this article (A).
3. Conclusion: C exists — Modus ponens, as we had the satisfying antecedent (A) we receive the consequent (C). Therefore, the article will be published (C).

You may have heard of a smattering of basic logical rules, possibly including the one used above. Here are some of the basic logical rules you may have already heard, perhaps employed by a pretentious wannabe philosopher, or maybe by a colleague:

Modus Ponens

• If A, then B
• A
• Therefore: B

Modus Tollens

• If A, then B
• Not B
• Therefore: not A

Hypothetical Syllogism

• If A, then B
• If B, then C
• Therefore: if A, then C

Whether we use the names of these logical rules of not, you can see from the above that these rules govern most of our arguments. In many cases, we completely take them for granted.

The time in which we no longer take these structures for granted is usually when someone commits a fallacy, or a common mistake of a logical argument. You may have also heard of some types of fallacies, such as:

Slippery Slope (Continuum Fallacy): A slippery slope argument is a fallacious form of a hypothetical syllogism in which the speaker asserts that an object or action (A) will result in unintended consequences (Z), roughly:

• If A, then B
• If B, then C
• …
• If Y, then Z
• Therefore: if A, then Z

What makes this a fallacy is the omitting of logical connections between (C) and (Y) above, inserting a connection where there is not one.

Straw Man: A straw man involves subtly replacing the original topic of argument (X) with a similar one (Y), and then subsequently focusing the argument on (Y) as if it were equivalent to (X). Roughly:

• X
• X is like Y
• If Y then Z
• Therefore: Z

When you’re arguing with someone, whether your discussion seeks to answer the meaning of life or if we should invest in Acme Corp., it is useful to understand these structures to determine if your conversational partner’s arguments are sound (“sound” = logically formulated and founded on correct premises).

As I said, symbolic logic is an accessible area of mathematics, clearly translating into argumentative applications. And, to be clear: logic is absolutely a form of mathematics, not a separate discipline. Yet, while it is helpful to trace your arguments using logic, you can learn these same pathways by studying math.

Mathematics is founded on proofs, whether you’re creating a symbolic logical arugment or taking upper-level calculus. When mathematicians explore new areas of math (yes, these exist), they use proofs to demonstrate their findings. Yet, even in cases where you are solving from point A to point B, you ‘show your work’ to demonstrate the process, just as you would cite logical rules to demonstrate how you arrive at conclusion B. Mathematics teaches you a system of thought that transcends applicability of literal rules, permeating how the user thinks about moving through an exploratory process. If you have a strong understanding of math, you will be able to see logical connections and fallacies without defining the precise rules as above.

Hopefully by now you understand why math is about more than addition and multiplication, or even those infamous imaginary numbers. Studying mathematics (or symbolic logic) can help you address and reformulate arguments across all disciplines, providing the structure for clarity and precision.*

In our final section, we will explore why studying and understanding these systems of thought — whether through explicitly applying logical rules to your arguments or by understanding mathematical structures — is an overlooked way to gain an edge in business.

Ready to keep going? Read part 3.

* One discipline that does this well, and is often the companion field of math, is philosophy. Philosophy uses a number of deductive methodologies reflecting the logical rules above. It also is known for using another methodology called thought experimentation, which in many circumstances deviate from deductive logic (although it sometimes uses deductive logic as well), using an approximation of inductive logic to determine what premises might necessitate a given outcome. For more on how business people can apply thought experiment methodology to their work, check out my article “Experimenting with Thought”.