What does math teach students who won't need university-level math, that Logic can't?

Question

Sources: 1 by Tim Gowers. 2 by Marcus du Sautoy.

This question assumes that pre-calculus, probability & statistics ought still be taught in high school, and involves only those who won't need university-level math. Can studying Logic also teach you the non-quantitative benefits cited below (e.g. thinking, pattern searching, problem solving)? If not, why are they exclusive to studying math?

[1.]

Mathematics should be a tool for increasing one’s thinking power but for many children it is just a set of rather pointless rules for manipulating symbols.
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It is therefore good for the health of a country if its population has high standards of mathematical literacy: without it, people are swayed by incorrect arguments, make bad decisions and are happy to vote for politicians who make bad decisions on their behalf.

[2.]

But why should maths be privileged above learning a foreign language or history? Does everyone need to know what a cosine is if the UK is to have a brighter future? Does the success of our economy depend on every citizen feeling confident factorising a quadratic equation? It may come as a surprise to you that I don’t think so, but I’m still a big believer in teaching maths to 18. What will be important is making sure that the maths we expose students to is both relevant to their future and the future of our country.

What many are not aware of is that maths is so much more than the technical cogs that currently form the backbone of the curriculum. It is about pattern searching, extended analytical and logical thinking, problem solving. I am just embarking on making a new programme for the BBC about the beauty of algorithms. Many of the best algorithms contain no numbers or equations at all, but are full of mathematical thinking. And it is those algorithms that are creating efficient approaches to a whole range of business solutions, from the distribution of goods from supermarket warehouses to decisions about flight schedules at Heathrow airport.

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What about those humanities students or creative artists or vocational students who might argue that they will never need more maths? I believe that even these students, if exposed to the right curriculum, will recognise the benefits of more maths. I am doing an event with the Booker-winning novelist Ben Okri at the Hay festival next month about the connections between mathematical proof and literary narrative. As a novelist, Okri is the first to recognise the importance of a logically consistent narrative to the success of a novel – but also the wonderful benefit that a mathematical sensitivity to pattern and structure can give novelists as they create a narrative arch. From musical composition to carpentry, from street art to journalism, a mathematical mindset potentially gives one an edge.

This Reddit comment affirms that math is logic:

And mathematics is philosophy, it's just a lot less "wordy" and rather brought to the meta level. I mean just look how complicated you have to explain a valid and a sound argument and how nicely you can calculate with them in the boolean framework.

Answer 1

The question is about students who "won't need university-level math", but a prediction about the future needs of the student isn't a fact. It's a guess. What if every path pursued by a student reaches a dead-end of failure, and the student begins to reconsider possible goals, and all of the most attractive and feasible goals seem to require university-level maths? What if a given student discovers in future that university-level maths are needed for what the student -- after a re-evaluation of goals -- ardently hopes to achieve?

If we are really talking about having a limited amount of time, and avoiding mathematics simply because there isn't enough time available for the student to study everything that the student wants to study, then there may be some ways to increase the amount of logic studied, and reduce the amount of math studied. However, is that the situation?

Another possibility is that the mathematics simply gets difficult, and the student wants to avoid those difficulties. For example, if the student is confused by a statement that begins with three quantifiers (such as "for every epsilon greater than zero, there exists delta greater than zero, such that for every real number x, ...") ... or if the distinction between ordinary continuity and uniform continuity of a function mapping the real numbers to the real numbers is too subtle for the student to appreciate, ... then the student is unlikely to proceed very far in the study of logic.

If, like a judge, you hope to recuse yourself from matters where there may be a conflict of interest, but you take things to the extreme of neither endorsing nor casting doubt upon any assertions whatsoever, then you may be free to concentrate on questions of pure logic. However, an interesting or important problem in pure logic is likely to have its origins in aspects of reality that go beyond pure logic. To create a problem in logic, you may generalize, simplify, and omit many particular details. However, you may find it valuable or essential to return to the source. That source may include messy and complicated mathematics, physics, astronomy, etc.

If you study history, then you will learn about Copernicus, Galileo, Kepler, and Newton. Galileo used perfect circles in his model of the solar system, but Kepler used ellipses. Ellipses are conic sections. We are back to mathematics.

How did people compute before there were electronic computers? The word "computer" used to name an occupation (like "baker", "tailor", or "sailor"). People who worked as computers used books including tables of values of logarithms, and the purpose of those logarithm values was simply to allow for efficient multiplication.

"Does everyone need to know what a cosine is if the UK is to have a brighter future?"

In many cases, it's easier to create an equation for a curve in 3-dimensional space that relates x, y, and z, if we begin with a parametric equation that involves not only x, y, and z, but also an additional parameter such as the variable t. Now, if we omit z, and consider a curve in two-dimensional space, then we can describe a circle as follows: it's the set of points (cosine(t), sine(t)) as t takes on values from zero to twice pi.

There's a bit more to it than that. We define cosine so that all real numbers are in its domain, and so that it's a periodic function. However, the main idea is that cosine(t) is simply the horizontal coordinate value, and sine(t) is the vertical coordinate value for a point on a circle.

Here's a new question:

"Does everyone in the UK need to know what width and height are if the UK is to have a brighter future?"

Maybe you have heard that Donald Trump is six feet tall and has a waist size of 36 inches or three feet, but if it's difficult to remember which direction we measure to discover what is called "height" and if it's difficult to remember what the word "waist" means, then some people might get those dimensions mixed up and announce that Donald Trump is three feet tall and has a waist size of 72 inches or six feet.

"for many children it is just a set of rather pointless rules for manipulating symbols"

Yes, but that's not a problem with the subject area itself. That's a problem with what is taught in schools, how it is taught, and the selection process used to determine who will be teaching math. Now we're getting into the topic of political policy, a topic quite far from pure logic!