I don't like this line of argument. It applies to many things, many of which we'd laugh at for suggesting.
Most people don't need to know how to read. Most people don't need to know how to add. Most people don't need to know how to use a computer. The foolishness of these statements are all subjective and based on what one believes one "needs". Yet, I have no doubt all of these things can improve peoples lives.
I'd argue the same with calculus. While I don't compute derivatives and integrals every day[0], I certainly use calculus every day. That likely sounds weird, but it is only because one thinks that math and computation is the same. When I drive I use calculus as I'm thinking about my rates of change, not only my velocity. Understanding different easing functions[1] I am able to create a smoother ride, be safer, drive faster, and save fuel. All at the same time!
The magic of the rigor is often lost, but the magic is abstraction. That's what we've done here with the car example. I don't need to compute numbers to "do math", I only need to have an abstract formulation. To understand that multiple variables are involved and there are relationships between them, and understanding that there are concepts like a rate of change, the rate of change of the rate of change, and even the rate of change of the rate of change of the rate of change! (the jerk!)[2].
That's still math. It may not be as rigorous, but a rigorous foundation gives you a greater ability to be less rigorous at times and take advantage of the lessons.
So yes, most people "don't need calculus" but learning it can give them a lot of power in how to think. This is true for much of mathematics. You may argue that this is not how it is taught, but with that I'll agree. The inefficiency of how it is taught is orthogonal to the utility of its lessons.
[0] Is a physicist not doing math just when they do symbol manipulation? I can tell you with great confidence, and experience, that much of their job is doing math without the use of numbers. It is about deriving formulations. Relationship!
I get the argument you're making but that's a bit like saying cavemen used to do calculus as they hunt, which is a valid way of looking at this maybe but they didn't really "use calculus" just intuition. Simillarly, when learning calculus, most people do not do so at a driving course, they do it in the classroom.
If you're willing to stretch the definition of what "using" maths is then it can apply to everything and that devalues the concept as a whole. I'm not on the toilet, I'm doing calculus!
I understand that interpretation but it's different from what I meant.
The difference may be in two different cavemen. One throws his spear on intuition alone. The other is thinking about the speed he throws, how the animal moves, the wind, and so on. There is a formulation, though not as robust as you'd see in a physics book.
> the definition of what "using" maths is then it can apply to everything
In a sense yes.
Math is a language, or more accurately a class of languages. If you're formulating your toilet activities, then it might be math. But as you might gather, there's nuance here.
I quoted Poincaré in another comment but I'll repeat here as I think it may help reduce confusion (though may add more)
Math is not the study of numbers, but the relationships between them.
Or as the category people say "the study of dots and arrows". Anything can be a dot, but you need the arrows
Yeah, I do understand your point of view. I'm just doubting if it applies universally, like you may superimpose that assumption on the thinking caveman, but is the thinking caveman really doing the same?
Yes, technique is one thing, but being really good at throwing spears doesn't make you really good at math, is my argument. And most people will encounter maths in a formal setting while lacking the broader perspective that everything is technically "math".
Yet, we need to see the argument from the common person's view, if we're talking about calculus and learning in the traditional sense. The view you stated is quite esoteric and doesn't generalize well in this setting imo.
It's like a musician saying they see music in every action, but to most non-musicians (even if the stated thing is kind of true) that doesn't make a lot of sense etc.
> but is the thinking caveman really doing the same?
Are you projecting a continuous space onto a binary one? You'll need to be careful about your threshold and I'm pretty sure it'll just make everything I said complete nonsense. If you must use a discrete space then allocate enough bins to recognize that I clearly stated there's a wide range of rigor. Obviously the caveman example is on the very low end of this.
> It's like a musician saying they see music in every action, but to most non-musicians (even if the stated thing is kind of true) that doesn't make a lot of sense etc.
Exactly. So ask why the musician, who is certainly more expert than the non-musician has a wider range? They have expertise in the matter, are you going to just ignore that simply because you do not understand? Or are you going to try to understand?
The musician, like the mathematician, understands that every sound is musical. If you want to see this in action it's quite enlightening[0]. I'm glad you brought up that comparison because I think it can help you understand what I really mean. There is depth here. Every human has access to the sounds but the training is needed to put them together and make these formulations. Benn here isn't exactly being formal writing his music using a keyboard and formalizing it down to musical notes on a sheet (though this is something I know he is capable of).
But maybe I should have quoted Picasso instead of Poincaré
Learn the rules like a pro, so you can break them like an artist.
His abstract nature to a novice looks like something they could do (Jackson Pollocks is a common example) but he would have told you he couldn't have done this without first mastery of the formal art first.
I know this is confusing and I wish I could explain it better. But at least we can see that regardless of the field of expertise we find similar trains of thought. Maybe a bridge can be created by leveraging your own domain of expertise
Maybe I can put it this way: gibberish is more intelligible when crafted by someone who can already speak.
Most people don't need to know how to read. Most people don't need to know how to add. Most people don't need to know how to use a computer. The foolishness of these statements are all subjective and based on what one believes one "needs". Yet, I have no doubt all of these things can improve peoples lives.
I'd argue the same with calculus. While I don't compute derivatives and integrals every day[0], I certainly use calculus every day. That likely sounds weird, but it is only because one thinks that math and computation is the same. When I drive I use calculus as I'm thinking about my rates of change, not only my velocity. Understanding different easing functions[1] I am able to create a smoother ride, be safer, drive faster, and save fuel. All at the same time!
The magic of the rigor is often lost, but the magic is abstraction. That's what we've done here with the car example. I don't need to compute numbers to "do math", I only need to have an abstract formulation. To understand that multiple variables are involved and there are relationships between them, and understanding that there are concepts like a rate of change, the rate of change of the rate of change, and even the rate of change of the rate of change of the rate of change! (the jerk!)[2].
That's still math. It may not be as rigorous, but a rigorous foundation gives you a greater ability to be less rigorous at times and take advantage of the lessons.
So yes, most people "don't need calculus" but learning it can give them a lot of power in how to think. This is true for much of mathematics. You may argue that this is not how it is taught, but with that I'll agree. The inefficiency of how it is taught is orthogonal to the utility of its lessons.
[0] Is a physicist not doing math just when they do symbol manipulation? I can tell you with great confidence, and experience, that much of their job is doing math without the use of numbers. It is about deriving formulations. Relationship!
[1] https://easings.net/
[2] https://en.wikipedia.org/wiki/Jerk_%28physics%29