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

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

ath92 5y ago
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.
- xNeil 5y ago
  Surprisingly for me, it's the other way around! Equations build intuition in my case, although I'm not sure why.
  - thomasahle 5y ago
    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.
    - kortex 5y ago
      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.
lelanthran 5y ago
>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"
- anon_tor_12345 5y ago
  >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.
  - ska 5y ago
    > 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.
    - anon_tor_12345 5y ago
      >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).
seoaeu 5y ago
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.

ath92 5y ago

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.

xNeil 5y ago
Surprisingly for me, it's the other way around! Equations build intuition in my case, although I'm not sure why.
- thomasahle 5y ago
  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.
  - kortex 5y ago
    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.

lelanthran 5y ago

>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"

anon_tor_12345 5y ago
>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.
- ska 5y ago
  > 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.
  - anon_tor_12345 5y ago
    >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).

seoaeu 5y ago

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.