Hacker News Reader

Comment by etamponi | Hacker News Reader

From the beginning of the article:

> The Fourier Transform is one of deepest insights ever made. Unfortunately, the meaning is buried within dense equations

The post does a great job at explaining what the Fourier transform means, but I am always skeptical of sentences like this. It's a pattern found in many other posts too: "unfortunately" this concept is expressed in math form, let's try to solve this problem.

I find that it's exactly the other way around. Math and math notation is dismissed too early and too easily even in engineering studies, resulting in shoddy and imprecise definitions and understandings.

> The Fourier Transform is one of deepest insights ever made. Fortunately, the meaning is engraved in dense equations

View on HN

>It's a pattern found in many other posts too: "unfortunately" this concept is expressed in math form, let's try to solve this problem.

Some things are less intuitive because of the dense equations. You'd have to be slightly insane to say that "x^2 = y^2 + z^2" is a more intuitive way to explain what a circle is than a picture of a circle.

> Fortunately, the meaning is engraved in dense equations

"I am very intelligent"

>You'd have to be slightly insane

yes and this particular flavor of insanity has a name: pedantry.

>the meaning is engraved in dense equations

what's hilarious is that middling tier mathematically literate people worship mathematical formalism while truly mathematically literate people (i.e. mathematicians) eschew it because they understand that mathematics can't be formalized but also because it's natural language whose value is communication not codification. even here - ask someone to explain to you rigorously the definition of the fourier transform wrt dirac deltas. again the middling tier mathematically literate will assume that writing down an integral against a dirac delta is rigorous simply because you can write it down completely forgetting that the dirac delta is a tempered distribution not a function and thus that integral means something completely different. ultimately it is the case that the definition is consistent with a set of agreed upon definitions but the symbology itself is just a pun.

> the dirac delta is a tempered distribution not a function

Nah, it's a measure :)

Joking aside your point is a good one. Notation is a necessary but flawed tool to communicate the details with minimal confusion.

>Nah, it's a measure :)

lol tell me you're a physicist without telling me you're a physicist :)

>with minimal confusion.

i think this the key thing that people that do a little math but not an immense amount don't understand: notation serves the same purpose as short hand in any other speech/communication. locally it's crystal clear, where local can be as small as that paper, that subsection, or just that paragraph (cf "the distinction is clear from context" lolol). i've written papers myself where i switch notations midstream for efficiency's sake (try defining tensors without bases but then try manipulating them without index notation...) but taking notation for granted is just as likely to get you into trouble as taking an API's promises for granted (in fact they're exactly the same things - leaky abstractions).

the value of [mathematical] language is not only communication, but also reasoning. It is remarkable, and sometimes unfortunate, how much mathematical reasoning you can do without knowing what you're doing, by pattern matching. Notation doesn't just allow you to communicate. It also allows you to manipulate e.g. truths that you know into truths that you do not yet know. In doing this abstract manipulation, you postpone (when necessary, and hopefully not indefinitely) thinking about semantics, while acting only syntactically.

A lot of people say something like "math is a universal language", and it sounds like we both agree that's nonsense. But I think we disagree on the extent to which math is a natural language. Your comment does not, in my opinion, recognize the symbolic manipulation (which doesn't have to be mostly-linear sequences of symbols, but can be e.g. diagrams too) that make math math.

You're taking this to a really uncharitable extreme, but yay you can laugh at the "hilarious" "middling tier mathematically literate people," I guess.

> You'd have to be slightly insane to say that "x^2 = y^2 + z^2" is a more intuitive way to explain what a circle is than a picture of a circle.

I would expect any freshmen STEM student to recognize sqrt(x^2+y^2) as the distance of an (x,y) point to the origin. And the equation x^2+y^2=r^2 says this quantity must be fixed.

Now I wouldn't expect a student to be able to write the previous two sentences so clearly.

This gets to the heart of the paradox of learning. How to see past the clutter if you don't know what you're looking for?

The human brain is lazy and prone to skipping logical steps, as reflected in the variation in natural language (and student's writing skills). Therefore learning a natural language does not demand simplicity up front.. unlike the up-front rigor required to learn mathematics notation or a programming language.

>How to see past the clutter if you don't know what you're looking for?

the picture of the circle isn't clutter - in a very precise sense it is the definition (platonic ideal). the algebraic definition "x^2 = y^2 + z^2" clearly isn't the definition because it's missing things (which field is this polynomial over?). there are many other definitions (another is R/Z).

>mathematics notation or a programming language

notation isn't a formal language - there is no formal grammar written down somewhere and in fact there can be none (cf godel). whether you realize it or not it's a natural language that evolves, has slang, and is spoken to various degrees of competency by many successful mathematicians.

> the picture of the circle isn't clutter - in a very precise sense it is the definition (platonic ideal)

Which precise sense? I get that language is slippery, but I can't follow you if you get philosophical, bring up fields (maybe the polynomial is defined over a ring!) while simultaneously stating that your favorite definition is The One True Way (TM).

How does one use the platonic ideal to distinguish a circle from an ellipse?

> whether you realize it or not it's a natural language that evolves

Yes, I realize.

My point was that the day-to-day usage of natural language tolerates a lot of ambiguity. Whereas the classroom usage of mathematical/programming notation does not. It's hard to be informal and "break the rules" as a beginner while retaining precision. Because the web of potential associations hidden in a statement (the "clutter") hasn't yet been pruned down by experience.

Writing is hard.

>Which precise sense?

in the sense that "mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging"

https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Plat...

>How does one use the platonic ideal to distinguish a circle from an ellipse?

entailed in the picture "definition" is constant curvature.

>while simultaneously stating that your favorite definition is The One True Way (TM).

that's not what i wrote.

>Whereas the classroom usage of mathematical/programming notation does not.

i won't speak to programming notation but every upper level or grad math class i've had has been at times extremely informal in its use of notation. take an integral on any chalkboard in any class and you're almost certain to have these questions without listening to the lecture itself (i.e. it's omitted from the notation):

1 what is the domain? R or C or something really exotic

2 is it over a closed contour/boundary or open?

3 what's the metric? i.e. are we actually integrating forms?

4 is it even an integral at all not just shorthand for solving a differential equation (e.g. stochastic integral)

> in the sense that "mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging"

oh THAT sense, why didn't you just say so?

but seriously what does that sense have to do with a picture of a circle? how would one use the picture of a circle to distinguish from an ellipse?

edit: nobody ever checked if i knew diddly squat about "real actual academic platonism v formalism" before peforming mathematical exposition in front of numerous classrooms in at least two countries, and surely i'm not going to hold back while arguing on the internet!

>oh THAT sense, why didn't you just say so?

i mean if you're not familiar with the real actual academic platonism v formalism debate maybe you shouldn't pontificate on what's better or worse re mathematical exposition?

> Some things are less intuitive because of the dense equations. You'd have to be slightly insane to say that "x^2 = y^2 + z^2" is a more intuitive way to explain what a circle is than a picture of a circle.

It's the difference between qualitative and quantitative descriptions. Which one captures the full depth of the concept? Neither on its own; you need to contemplate both at the same time to capture that depth ("range" may be a better word). There is information hiding in the interplay between the two that is missed when considering them in isolation.

I don't believe formalism is all we need, but it is necessary if the problem at hand requires more than just intuition. Conversely, formalism without qualitative understanding is opaque and sterile. Together, they combine operational ability and simple intuition into a higher form of intuition (operational intuition), the most desirable kind.

As a compromise between you two:

> Fortunately, the process is engraved in dense equations. Unfortunately, the dense equations make it difficult to learn why we're following this process.

For most people, having only the equations means they cannot build an intuition for what a fourier transform actually is, because they don't have a solid enough grasp of math notation to easily make sense of what the dense equations actually represent.

For those people it's very helpful to first get a sense of what it all means (even if it means that initial mental model is somewhat imprecise), because it makes understanding the equations easier.

Surprisingly for me, it's the other way around! Equations build intuition in my case, although I'm not sure why.

My experience is that visuals help people _feel_ like they understand the concept, but the equations are necessary to get an _operational_ knowledge.

It's all well and good to be able to talk about the "deepest human knowledge" at a dinner party, but it doesn't help you solve actual problems using the fourier transform.

My brain works the opposite. Equations give me something I can work with but feels like voodoo, and I only get operational knowledge once I can visualize it in some way.

But "dense" in this context literally means "difficult to understand". If you are trying to build intuition, then that is pretty much by definition a bad thing.

Reply on news.ycombinator.com