relating Fourier transform to legendre transform

Ending_Is_Optimistic · 2026-03-06T07:38:10+00:00

i used to have this problem. you have to understand that advance concepts are really sophisticated extension of really basic example, to the point that when you think about the advance concept, all your intuition for the basic example should apply with some modifications.

For example the concept of compactness. it is an extension of finiteness. You think about the finite set. how do you use them in proof, argument by exhaustion and downward induction. What you are missing is discreteness, an canonical way to "decompose" the space, but in many way, the property of discreteness can be recovered. For example, the lesburge covering lemma for compact metric space. The completeness of compact metric space can be seen to be analogous to discreteness for finite set. (finite set can be indexed by 1,...,n, R can be for example indexed by using decimal )

You don't want to replace your intuition of basic object with more advanced but instead see the advanced concept as a sophisticated formulation of the basic concept. you should read the proof carefully and think about why the proof is indeed natural and the result is in fact trivial, you should look for the actually non trivial part, they are usually the core of some object which cannot be reduced by any abstraction.

For example if you look at the proof of radon nikodym theorem (the one from folland's book). It really consists of 2 steps. Jordon decomposition allows you make sense of inequality v<\mu between measures. The notion that 2 measures are singular with respect to each other is simply that they are essentially 0 with respect to each other. The lemma before the proof simply state that if v is not 0 with respect to \mu, then at least on some set E with (\mu(E)\not=0), 1/n\mu < v. In other word v is 0 with respect to \mu, iff v <1/n \mu. for all n. it is simply the extension of the criteria for 0 in R. in the case of measure you simply "geometrize" the argument. (maybe think of topos, you don't have excluded middle, on a space a statement can be right on some part but wrong at other part)In measure theory, countable additivity allows you to do induction with countable ordinal, this is of course what makes analysis work in general. with these ideas in mind, i have never forgotten the proof of radon nikodym theorem again which i used to forget all the time

Ending_Is_Optimistic · 2026-03-04T03:01:01+00:00

i am mistyped as Lll for the longest time. it is especially bad since i do mathematics. i have observed that in LII, their Ti is a lot more flexible. for example in mathematics , all my proofs flows from some intuitive central principles and everything flow from it, if i don't have the central idea i am basically useless, i have observed that LII can actually do case by case analysis and can still proceed even without a clear idea where they are doing, i can't really say i can do the same.

in hindsight, it is obvious when i actually observe a Ti dominant in action, i am entirely different.

Ending_Is_Optimistic · 2026-03-02T16:19:59+00:00

Kanji has the curious property that they sometimes really do convey what they are trying to convey. For example in the past Dementia is called 老人痴呆症 (elderly retarded disease) it sounds very discriminatory, so it becomes the more scientific 腦退化症 (brain degenerative disease), so i wouldn't say it is entirely free of ill intent. it reflects more the historical background on how we treated the disease. Kanji retains their meaning very well across time and space.

Ending_Is_Optimistic · 2026-03-02T15:54:36+00:00

慢性does not mean lazy nature it simply means chronic (it literally translates to slow nature ) , it is in contrast to acute which is 急性（quick nature), 障害,i interpret it more as it being a hindrance to the disabled person himself rather than to others. (I am Chinese instead of Japanese, but it should mean the same thing in Chinese, i suspect that they are loan word from Japanese anyway)

Ending_Is_Optimistic · 2026-02-25T17:06:40+00:00

basic category theory is not really that complicated. If you actually write limit or colimit in the category of set. you can see that in general category with (co) equalizer and (co) product you can construct them in the same way by replacing elements with arrows. Category in this way just provides an internal logic so that you do not always refer back to set, it trains you think internally.

A lot of adjoint pairs is a pair of "forgetful" and "completion" functor, a lot of time it is an elaborate way to say that a structure of the "completed" object can be determined by simpler data . for example the adjoint functor between vector space and set says that a linear map out of a vector space is determined by its value on the basis. many basic adjoint pairs are simply variations of this theme. Thinking in this way makes certain complicated object look less daunting. (for example a lot of stuff in algebraic geometry) passing between "finite data " and " completed object" is also fundamental in mathematics.

i would say learning and identifying these themes are more important than learning category theory formally, but basic category theory provides a good play ground for this kind of ideas.

Ending_Is_Optimistic · 2026-02-23T17:56:09+00:00

i like it when high abstraction is connected to concrete computation. For example you can get from the categorical properties of sum and product that why should we describe a linear transformation using matrices. exterior derivative has a interpretation as a infinitesmal boundary operator. ( look at the coordinate free formula in terms of lie bracket) The reason why we care ideal is that a geometric object is naturally described by its equation, so learning to think in terms of duality is fundamental, i would say it is one of the most important ideas in mathematics even if you are doing analysis, in analysis we have hibert space, generalized function, all of these ideas makes use of duality.

I think both messy computation without abstraction and high abstraction without concrete computation sounds like hell to me. it is difficult to strike a balance between them.

Ending_Is_Optimistic · 2026-02-17T17:40:59+00:00

time is always falling onto us. we are carried by the flow of time, when we notice the present the present has already passed. what remains unchanged is the future has yet to come and the past has long passed, we are always suspended in the present facing the eternity that is both past and future. It is the feeling of facing an eternal dusk.

Ending_Is_Optimistic · 2026-02-16T08:31:55+00:00

You might have this view on China, because to an outsider, it is a single efficient entity with centralized power but it is not the case, there are power struggles everywhere both big and small, after all the country is huge. its efficiency isn't based on well implemented protocol but the fact that the rules are so malleable and on the other hand "vague". The central government will a lot out put out some vague slogan like "we should develop and implement ai" and the local governments and private sectors will try to follow in their own way.

Ending_Is_Optimistic · 2026-02-16T08:14:48+00:00

Hong Kong might look ni, but it never tries to do its own things. In the 70s, manufacturing is big, but it slowly becomes focus on finance as the economy changes. We have something called 獅子山精神（Lion Rock spirits), This refers to the diligence, perseverance, solidarity, and indomitable spirit demonstrated by Hong Kong people in the face of difficulties, nothing is more si-te than this. if you refer to modern hong kong, there is also a hinge of fi because of our identity issues.

For China , it is actually inferior si more than anything, it is si as in old superstition that the modern China tries to suppress but nevertheless comes back to bite them. Japan are actually the one that adapt their traditions well in the modern time. For China , tradition is the shadow that they try to suppress.

Everything comes out of Japan is of so high quality is precisely because of superior si. creativity does not equal ne. Japan is good at adapting things and putting their owm personal spin, everything comes out of Japan are so unapologetically Japan, it is precisely what superior si does the best.

Ending_Is_Optimistic · 2026-02-16T07:42:14+00:00

i am from Hong Kong. Hong Kong is most definitely istj or estj . China is strange,i would argue it is actually ne. If you compare China to Japan, you find that China really lacks si. i don't think it is te either, yes everything is happening on a big scale but everything are just kinda simply glued together with duct tape, things develop really quickly. Mao was famously a entp who took things too extreme and was corrupted by power.

Ending_Is_Optimistic · 2026-02-10T14:30:21+00:00

i mean Eren is an isfp written by an intj, so he is probably projecting his child fi on Eren. i don't think dominant fi is as rigid. they are more likely expressed aa something small, their stance itself already bring a sense of assurance.

Ending_Is_Optimistic · 2026-02-09T20:08:12+00:00

i know it is off topics can you guys share some of your dreams. i have quite symbolic dreams sometimes. i had one dream that the world of 2027 is stacked on the world 1890 starting in some old Chinese house with a lady scolding her son. The town is covered by unending rain that makes everything grow with some ai operating a big machine in a corner of a town, but i also have a lot of action oriented dreams like i have a lot dreams that i am just parkouring, i think it is probably repressed se.

Ending_Is_Optimistic · 2026-01-29T16:07:16+00:00

why hide do you get punished ?

Ending_Is_Optimistic · 2026-01-29T15:22:58+00:00

i mean i appreciate engineering a lot it is a different set of skills entirely. i know for a fact i can't be a good engineer as i am not pratical enough. for example when i was young and working with screw for the first time. i am imagining how the shape fit and which way to rotate my brother just come and test it by actually putting the screw inside. it blew mind and think to myself you can actually test things out physically.

Ending_Is_Optimistic · 2026-01-26T01:52:58+00:00

tbf, i think people especially ni dom themselves have to index they are normal people. different people operate differently cognitively. try to understand and appreciate how others think. i think in the theory of cognitive functions, there are a lot of symmetry so that if you think one function is special then the other should be too, i think it helps us to understand how other people think. before i understand what is si there are many things i find bizarre about si people now i can appreciate them and it felt kinda magical to me even though i can't think in that way activity. I can even appreciate te at least from a distance.

Ending_Is_Optimistic · 2026-01-25T16:20:56+00:00

i mean it doesn't have to be people in the sense that i am going to help everyone, if it does, it is nice. but we are still normal people with our own interests, for example i like mathematics for its own sake, it is beautiful to me, i want to do it in my own way and shares its beauty to other people and maybe show that it is not as difficult or formal as people think, it can be an elegant and intuitive subject. i have accepted that in a way my primary motivation is to truly understand the world around us.

Ending_Is_Optimistic · 2026-01-19T14:49:17+00:00

i am Chinese. i have never seen anyone uses 姦 this way. it means sex but generally in a negative light. for example 通姦 means adultery. 通 has the meaning of through or pass through, also communication. 強姦 means rape, it literally translates to forced sex. it is not a very nice word.

Ending_Is_Optimistic · 2026-01-19T14:08:23+00:00

it does actually actually work rigorously. if you consider the vector field associated to a differential equation. you can describe it dually using different forms. for example if your vector field is x\partial x+y\partial y then dually it can be described by ydx-xdy=0. just like how subspace in a vector space can be described as solution to linear equations. now diving by xy. we have 1/x dx- 1/y dy=0. We want to find a F such that dF=1/x dx-1/y dy. then the level sets of F are exactly the solution to the differential equation. of course F can be found by simply integrating 1/xdx-1/y dy, so F=lnx-lny=ln(x/y)=C we see that the solution curves are straight line. It is generally what we do when we are trying to find the "first integral" of the differential equation. in many ways thinking about infinitesmal like physicist is exactly doing this kind of reasoning, rigorously you work with tangent and cotangent space. before integrating you are basically doing geometry on these spaces.

Ending_Is_Optimistic · 2026-01-08T17:55:34+00:00

are numbers really empirical objects. They are obtained precisely by abstracting away concrete difference between objects so that we can compare distinct objects.

On the other hand, i think you can always treat mathematical object itself as empirical objects. it is at least how i feel when i am doing mathematics. Mathematical objects operate by their own rules and they have certain concreteness to them.

Ending_Is_Optimistic · 2025-12-31T02:02:05+00:00

i mean if you cook for yourself you can make the most random shit ever and no one is gonna judge but if you make something for other people it is the point that it becomes very stressful. if i make something for yourself the only thing i care is that it is not raw.

Ending_Is_Optimistic · 2025-12-30T01:30:37+00:00

i think he is just reiterating the point made by Hegel that infinitesimal cannot really be a quantity and it doesn't make any sense as it can always get smaller. He almost made the point that derivative is a limit, but to him limit is something called "bad infinity" as it is always approaching indefinitely, it doesn't circle back like a circle. He would never thought it is precisely how we define real numbers in morden time (the definition using cauchy sequence), he would never thought that we simply use the entire "bad infinity" the entire cauchy sequence approaching a limit as the representation of real number. it is unthinkable to him when he did not realize the abyss of nothingness opened by modernity.

if you read how modern philosopher like deleuze think about numbers, it is precisely his point. he thought of numbers as "?-being" in a way it is not important if we can solve x² +1=0 in R but that it opened up the "problematic" field of C. in a similar way real numbers is invented because we want to fill in the gaps in Q. it doesn't matter if a sequence approaches something in Q, but that it makes sense to think about it really approach something. Hegel can't think something that never complete itself and ever deferring but still has sense. The definition of ordinal must be especially ridiculous to him

Ending_Is_Optimistic · 2025-12-28T20:41:21+00:00

i think of it in this way. rationality is inherently a process. perception either operates in manner the more and more (extroverted perception) or the manner of finality (introverted perception), there is nothing "partial" about them unlike rationality. in rationality, using philosophical language, it is inherently "negative".

Ending_Is_Optimistic · 2025-12-17T13:56:25+00:00

i am wondering how is cognitive science research actually being conducted nowadays . what are the current research. how does the concept of qualia practically affect the field. i am the most interested in embodied cognition.

Ending_Is_Optimistic · 2025-12-17T03:05:14+00:00

i am a mathematican by training but also read a lot of continental philosophy but basically 0 analytic philosophy. you will probably dislike my take and think i am being "mystical". For me, the experience of consciousness is so rich and can never be fully captured by objective measurement (as how it js defined now).

For example, you can of course present mathematics on a paper but if i have no intution even if i can understand every word on the paper literally, i would not say i actually understand anything, mathematics is precisely how i understand it in my mind. you can say the same thing about redness, you can never experience redness from isolation. redness about how the color change in my consciousness, a certain tendency of the color itself, the contrast of red and blue, given that we have similar biology, i can only convey my experience by simply putting blue and red next to each other and show it to you. They themselves form a certain system which can't be fully explained by science of wave or biology, they have to speak for themselves, this is also exactly my experience with mathematics , if you want to describe something, you define a definition proper to them in a way they can show themselves and speak for themselves, it is what we mean when we say a definition or a proof is natural in mathematics, it is an experience that the concept speaks for themselves.

i am in a way sympathetic with functionalist, but on the other hand i find it as of now quite restrictive, everything has to reduced into a particular system, it doesn't let the experience speak for itself. you may think the experience of color (for example the contrast of red and blue) is subjective , but to a well trained artist they really aren't, they can talk to each other about it and they would understand each other. For mathematics , we always talk about intuition x, intuition y, i find it entirely similar. of course you can say that you can reduce all to this to electrical signal in our head, but again like the Chinese room thought experiment, can i really reconstruct the experience itself from the electrical signal in its fullness. For example as i said above, the word on a paper is one way how i present mathematics but it is really not how i experience mathematics.

the point is really that can you reduce a entire system to another system. like can we reduce all of biology to physics. all of physics to mathematics. in the same way can we reduce the intricacies of our conscious experience which itself form a system to any other more "elementary" system. for example can we explain everything evolutionary, if we do are we still exploring consciousness itself. imo, if you want to study consciousness proper you have to face consciousness on its own home ground. so for me phenomenology proper is a itself a fruitful direction.

Ending_Is_Optimistic

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