[adamnemecek]

> we need to learn 'the basics' and survive without tools to some degree

This is the wrong approach. There's no way you wouldn't understand math much better with these tools.

I'm hoping that in the future, math will be less about equations and symbols and more about graphing and being able to move around in the spaces described by the equations.

I would draw an analogy with a compiler. After using it for some time, your brain will take on the shape of the compiler and you'll write correct (lol is it ever tho) code without even having to compile it.

[btilly]

Both are true.

Clearly we are more productive with the tools. However it is very, very easy for people to see the tools as magic. At some point we need to actually understand what it is that we are doing. For which those equations and symbols are essential.

Yes, the computer can draw a pretty picture. Pretty pictures are helpful in conveying information. But they are a horrible way to understand inherently complex topics.

For example pictures are essential for conveying basic concepts in in multi-variable calculus. But you won't make much sense of the topic until you actually understand the three basic mathematical representations of a surface embedded in a higher dimensional space (function, level surface, and parametrized coordinates), how each connects to the tangent to the surface at a point (whether that tangent is a line, plane, or something higher dimensional). And you need to understand this in an n-dimensional way because that comes up, a lot.

So no, we won't lose equations and symbols. Ever. They are essential, and there is no possibility of real understanding without them.

[HillaryBriss]

> we won't lose equations and symbols. Ever. They are essential, and there is no possibility of real understanding without them.

i don't know.

sometimes i think these symbols and equations are just machinery created by deeply talented, deeply insightful people to help the rest of us arrive at correct statements and give us a little glimpse of a vast panorama of truth which their minds see intuitively, unaided by all the symbolic clutter.

[btilly]

Yes, and no.

Good mathematicians and scientists that I have personally known have a wide variety of styles of understanding and thinking. The first and most basic divide in mathematics is between people whose natural inclination is algebra versus analysis. The best at analysis seem to have a deep intuition like what you project. The best at algebra operate pretty directly with symbol manipulation.

[adamnemecek]

> So no, we won't lose equations and symbols. Ever. They are essential, and there is no possibility of real understanding without them.

I didn't say get rid of them. But I think that if you see a complex equation, you should be able to roughly imagine it in your head. Working only with symbols won't get you there.

[btilly]

But I think that if you see a complex equation, you should be able to roughly imagine it in your head. Working only with symbols won't get you there.

You're wrong. To take a real example, visualize G_{\mu \nu }+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }. I dare you. That's Einstein's field equations for General Relativity. I don't visualize a system of 2nd order partial differential equations involving many dimensions at each point in a 4-dimensional coordinate system. Do you?

Furthermore people differ on how visual they are. If you're someone who has to visualize things to understand them, you're hardly alone. Lots of people are like that. But then again there are people like me whose thinking is almost entirely non-visual. If I'm dealing with something abstract, not only do I not think visually, but a purely visual explanation doesn't really help me much.

[fnbr]

Hmmm. I would disagree with you. It is impossible to visualize the equations as that isn't possible to do in 3 dimensions, but it is possible to develop an intuition. After taking functional analysis, for instance, I began to develop an intuition for how function spaces work, and was able to visualize that to a certain extent (for a given definition of "visualize").

[adamnemecek]

You get it. Few people understand what's happening in every dimension at every point, but getting a feel is doable.

[posterboy]

The whole argument is that its not doable without the right tools and that visualizations are a very good tool for this.

[posterboy]

You might be thinking visually without even recognizing, because to call that kind of imagination visual (as in optical) is misleading. A Hydrogen 2 molecule has 6 degrees of freedom already, give that color and you have 8.

There was a story on HN about activity in the "visual" system of blind people. Here is a similar story google spit out: https://news.ycombinator.com/item?id=14720225

Commutative diagrams are a development that has applications in algebra, too. I'm currently reading Physics, Topology, Logic and Computation: A Rosetta Stone (J. C. Baez, M. Stay) https://news.ycombinator.com/item?id=12317525

[adamnemecek]

I mean this particular equation not really no but that might be because I haven't done much general relativity. But I can kinda visualize other somewhat complicated equations.

Sure but then you are giving an advantage to the more visual people no? Can't I just flip your argument?

Yes, symbols are useful but you should have an understand of what they are actually doing.

[btilly]

Sure but then you are giving an advantage to the more visual people no? Can't I just flip your argument?

Sure, you can flip the argument. You'll be wrong, but you just did it.

People naturally lean towards understanding using different methods. Each mode of thinking has strengths and weaknesses, and so do people. Some things are better understood visually. Other things not. Some things have the best way of understanding varying by person.

So yes, on some things you'll have an advantage over me because you're visual. On some things I'll have an advantage over you because I probably understand complex abstract relations more naturally than you do. And your insistence that everyone is best off trying to think like you do is completely misguided.

Anyways I've explained my position enough. If you don't wish to get it, you won't. I'll let other people take over the task.

[xiaoma]

> "I think that if you see a complex equation, you should be able to roughly imagine it in your head."

This breaks down in many cases. What if your equation involves complex numbers for each component? What if it's an eight dimensional space or perhaps a multi-hundred dimensional space for recommender system?

At some point, you have to be comfortable with more abstract representations.

[adamnemecek]

I think of complex numbers as just points/vectors.

As for multidimensional spaces, you might not be able to visualize all the dimensions at once but by doing them three dimensions at a time, you can go Pretty far.

You should check that the abstract representations still make sense intuitively.

[posterboy]

> Clearly we are more productive with the tools. However it is very, very easy for people to see the tools as magic. At some point we need to actually understand what it is that we are doing

And that's why computer science needs to be part of the math curriculum in schools asap.

[btilly]

Please, no.

Or at least not unless it is done well.

The problem is that we start with a reasonable idea like, "People should understand how the tools work." We make an obvious observation like, "Understanding computer science helps people understand how the tools work." Come to a conclusion like, "Computer science needs to be part of the math curriculum in schools asap." This turns into a mandate for educators who themselves have no understanding of computer science. Who then ask the question, "What does everyone need to know about computers?" Who then consult with what seem to them like appropriate experts. Soon you're hearing about how we'll have interactive computer science courses to guarantee that children have familiarity with how computers work. And to fanfare these are rolled out in schools.

Then you go and look at what is being done. A teacher who clearly doesn't actually understand how computers work has kids using a variety of interactive programs, ranging from Microsoft Word to animated presentations. They are calling that "computer science". No actual understanding is imparted. And the exercise reduces time available in the core curriculum that could have spent on things like quantitative reasoning. You know, stuff which ACTUALLY can lay the foundation for understanding how the tools work.

[posterboy]

> A strawman who clearly doesn't actually understand how computers work

Fixed that for you. I had good informatics teachers 10 years ago.

> quantitative reasoning

As in complexity of sorting functions? We did that, non-rigorously.

I just don't see how, say, meiosis and mitoses or vulkanism are any more mandatory to learn than e.g. Codd's normal forms or the workings of an ALU.

[btilly]

I described what I actually witnessed happening to my children in supposedly good California schools. It fits a pattern that has been frequently seen over a long time with a variety of technical subjects that were pushed to schools, starting with the New Math debacle back in the 50s and 60s.

I'm glad that you personally had a better experience. I believe that I described something closer to what we should expect to happen with such initiatives.

[reikonomusha]

I would claim that most folks who have a reasonable understanding of fundamental computer architecture and assembly are those who actively learned it. Folks who have just used a compiler for everything, in my experience, rarely actually develop the understanding innately. Compilers still have this "magic" element associated with them.

[adamnemecek]

It's not either or, it's a dance. You make a hypothesis in your head, verify it with a compiler. If result matches your hypothesis, move on. If not, investigate.

For me personally, I thought I kinda knew assembly until I started using https://godbolt.org when I realized that I really didn't know assembly. This site helped a lot of other people as well. Note that all it does is make the process of discovery faster. But it's a tool in the same category as a calculator.

[Chris2048]

Many vocations use computers, but should they all require learning about software, electronics etc?

If you don't need to "understand" the math, why risk opportunity cost learning it?

> I would draw an analogy with a compiler

staying "high level" is a good thing for some programmers. If every web dev had to dig into the low-level working of the browser, a lot less would get done.