E: by convention the square root symbol means the positive root which I always forget so I think your identity is true for non-negative x

Edited by Belphegor ()

This is a cool math YouTube channel

dimashq posted:https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAwThis is a cool math YouTube channel

The best!

graphicalUSSRinterface posted:i'm reading pinter's abstract algebra book right now

i'm picking it up quickly, also using another book called 'visual group theory' alongside it. it's kewl. my problem with math/programming though is that i love reading dry technical stuff too much and i'm skewed towards that rather than practice which i need to work at i guess

Lenin posted:Such is the first cause of “physical” idealism. The reactionary attempts are engendered by the very progress of science. The great successes achieved by natural science, the approach to elements of matter so homogeneous and simple that their laws of motion can be treated mathematically, encouraged the mathematicians to overlook matter. “Matter disappears,” only equations remain. In the new stage of development and apparently in a new manner, we get the old Kantian idea: reason prescribes laws to nature. Hermann Cohen, who, as we have seen, rejoices over the idealist spirit of the new physics, goes so far as to advocate the introduction of higher mathematics in the schools—in order to imbue high-school students with the spirit of idealism, which is being extinguished in our materialistic age (F. A. Lange, Geschichte des Materialismus, 5. Auflage, 1896, Bd. II, S. xlix). This, of course, is the ridiculous dream of a reactionary and, in fact, there is and can be nothing here but a temporary infatuation with idealism on the part of a small number of specialists. But what is highly characteristic is the way the drowning man clutches at a straw, the subtle means whereby representatives of the educated bourgeoisie artificially attempt to preserve, or to find a place for, the fideism which is engendered among the masses of the people by their ignorance and their downtrodden condition, and by the wild absurdities of capitalist contradictions.

https://www.marxists.org/archive/lenin/works/1908/mec/five8.htm

from https://link.springer.com/article/10.1007/BF01946586

toyotathon posted:but i don't have confidence in it, like i'd get w/ teacher feedback, from watching khan youtubes...

the feedback on problem sets, that labor, i feel is the real value of a $$$ education. the low-cost alternative model to the school, which comes w/ that feedback, isn't youtube lectures, it is student communal research. but i can't really find that. people want that sheepskin.

there are ways around this though; stackexchange is an amazing resource where everything you would have thought to ask has been asked pretty much, i got clarification on an obscure point there just yesterday where someone was confused exactly where i was. i don't think learning math has to be much different from learning anything else, i think thinking it is is ideology probably. you learn to reason about math and programming the way you become fluent in marxism, it's something you can have intuition and opinions about too, it's a language more than anything. also if you really need to i've found emailing professors gets you places, they like to chat and probably will email exhcnage with you happily

toyotathon posted:gUSSRi, any insights into abstract algebra? what's going on there

abstract algebra is all about symmetry and transformations/rearrangments of algebraic structures, i'm still focusing on groups. but a group is basically a set along with an operation that is associative (though not necessarily commutative), satsfies closure axioms, has an identity element and is invertible. it's a very vague concept that really only makes sense after looking at a bunch of examples and sort of solidifies itself in specific concrete applications. one example is modular arithmetic, like on a clock. one of the cool things i learned is about cycles of permutations and that every group is isometric to a group of permutations, which i promise is cool but hard for me to explain in a nontechnical way

edit: if i had to summarize abstract alegbra id say its about the way you can rearrange things while keeping them fundamentally the same.

Edited by graphicalUSSRinterface ()

lenochodek posted:

lenochodek posted:

this thread seems like your opportunity to make an argument instead of deploying "lenin" and "soviet" as buzzwords

graphicalUSSRinterface posted:every group is isometric to a group of permutations, which i promise is cool but hard for me to explain in a nontechnical way

Cayley's Theorem is beautiful, its proof too

It's not a practical result, but it's one of those results that's almost philosophical in what it says about the nature of groups

toyotathon posted:^^^ what's cayley's theorem? i never did group theory. i remember playing with cayley trees in undergrad but i assume that's unrelated

Let G be a group. Then G is isomorphic to a subgroup of a symmetric group. If G is finite, then G is isomorphic to a subgroup of S_{n}.

an isomorphism is a structure-preserving map between two groups; two groups that are isomorphic are actually the same underlying group, the elements are just labeled differently. for example, as groups, the set of permutations of the digits 1, 2, and 3 is the same as the set of rotations and reflections of a triangle. they're ontologically different but their algebraic structures are exactly the same.

Cayley's Theorem says that any group is either a subgroup of the infinite symmetric group or a subgroup of the symmetric group on n symbols, S_{n}.

but in a more philosophical or thematic sense what Cayley's Theorem says is that all groups are just abstract manifestations of symmetry

Edited by insta_gramsci ()

elemennop posted:yo cat admin, enable mathjax for the forums

cars posted:lenochodek posted:lenochodek posted:

this thread seems like your opportunity to make an argument instead of deploying "lenin" and "soviet" as buzzwords

I'm not sure if I have a point to make but here is my position in any case. (Also I'm not a mathematician, I'm active in the field of chemistry)

I think Fock truly stated his approach on how to handle interpretation of quantum mechanics in a very lucid and to the point way but I would put it like this:

in physics we could state the evolution of theories as follows

1. there is a contradiction between a prediction from the theory we use and the measurement of that which we predicted

2. a new speculative theory which theoretically enhances the old theory is proposed, and is empirically tested

3. in case the new speculative theory gives better predictions we adopt it until further enhancement is achieved

Described like that, it seems clear this is fully consistent with the dialectical materialist approach. However, when we look at the approach of some self-described Marxist-Leninists to the "New Physics" we see some issues arise. In my opinion most of those result from a confusion of on one hand incorrect idealist model interpretations with on the other hand models with weak predictive power. The latter is lethal for a theory but the former is an error of the interpreter and not of the theory/tool/model.

To give a concrete example, check out this snippet essay called "Marxism and Quantum Mechanics" from 1975:

The author objects to the Heisenberg Uncertainty Principle, which is a fundamental principle that indeed makes better and broader predictions than what preceded it. The objection seems to be not with the predictions, but with the fact that certain interpretations of this fundamental equation are idealist and/or incorrect, and indeed for didactic reasons sometimes an analogy with the observer effect is used, which is an interpretation which has no verifiable grounds in reality (and is actually imprecise and incorrect). In the end, because the new theory does not line up with the classical physical deterministic theory, the new theory is rejected in its useful form. The confusion that occurred here is that we are testing our new theories to the interpretation of the old theory instead of to material reality. Of course, this is an error, tautological, dogmatic, not empirical and at odds with dialectical materialism.

Some other examples of where I feel that similar errors are made is in case of the slow correction of Lysenkoism and in the soviet rejection of resonance structures because they are "Machist" ( see here (NB:article written from anti communist standpoint)), here too they reject an actual material enhancement (e.g. correct understanding of the abstraction that is resonance structures allows for more efficient planning of successful chemical synthesis of fine chemicals including medicines) on the grounds that some interpretations of the model are idealist, which is true, but does not affect the better result.

This ties into Lenin’s remarks on physical idealism and in my opinion his criticism is to an extent faulty. For example, the great mathematician Poincare worked on a broad range of topics in pure mathematics which includes original and fundamental studies in dynamical systems which at this stage in time have plenty of real-world applications. Essentially, his philosophy of science was at odds with Lenin's approach and yet were empirically more powerful e.g. his "conventionalist" stand that Euclidian geometry is not a priori true was essentially confirmed by later approaches to relativity and space-time. To call Poincare an empiriocritic does him injustice because because of some (imo) minor idealist aspects to his conventionalist philosophizing is of no importance compared to Poincare's actual body of mathematical work, which has many real life applications by now.

Lenin states:
*“A law of physics, properly speaking, is neither true nor false, but approximate"—writes Duhem (p. 274). And this “but” contains the beginning of the falsity, the beginning of the obliteration of the boundary between a scientific theory that approximately reflects the object, i.e., approaches objective truth, and an arbitrary, fantastic, or purely conventional theory, such as, for example, a religious theory or the theory of the game of chess.
*With some goodwill, Duhem's statement is actually correct - a law of physics is a tool and this tool has its limitations and is thus approximate. Enhancement of the tool results in enhancement of the result. When we use this tool, perhaps we will interpret it somehow and maybe make some idealist assumptions (atoms are localized particles, molecular orbitals exist, such stuff) which are abstractions with no ground in measurable reality but which are still, although not real, useful to deploy the tool successfully. A religious theory of course has no predictive value which is better than alternative theories, so I don't think this is a valid angle of criticism to Duhem. But what about "the theory of the game of chess" or more broadly, abstractions which are mathematically consistent but not at all connected with the matter of physical reality?

I would call those theories "internally true abstractions" and in fact the theoretical toolkit which we can use when we want to build the tools that learn us how to engage with reality properly. This set of internally true abstractions essentially contains all mathematics without applications. Concretely, we can look at the famous example of "complex numbers" - for many years since their formal introduction in the 16th century they were viewed as senseless folly, fictions for bored math people until the work of Euler and Gauss and by now they have many concrete applications via quantum mechanics, electromagnetism etc. When we present theoretical work as reality, we become idealists. But when we apply these abstractions for concrete enhancements of theories for grasping material reality, they are objectively useful. With this empirical knowledge in hand, we know that developing and working on "internally true abstractions" on the long run provides us with ingredients to develop our concrete tools and material theories and this evidence should convince us to entertain these abstractions and not reject them a priori on philosophical grounds.

Of course, Lenin himself does not argue for that explicitly and he in fact welcomed the fusion of mathematics and the new physics of his time. That said, I think based on my statements Lenin would probably classify me as a "physical" idealist.

toyotathon posted:not really math strictly but i'm trying to learn kinematics, and drew this to convince myself of an explanation for Exact Constraint

exact constraint is a graphical kinematic method that reasons, there are 6 degrees of freedom in this universe, and so to make a body go in N degrees of freedom, it must be constrained in 6 - N.

"When an object touches a surface, the object begins to orient itself. Let me use this example: Imagine four balls on the edge of a cliff. Say a first ball starts at the top of a cliff and rolls down it. After a time delta-t, the formerly first ball becomes the second, a second rolls down, and then a third, and then a fourth rolls down the cliff."

bozhe moi

Susskind isn't perfect, but I do genuinely feel like I get the math of what's going on now, and a visceral understanding of the fact that it is just

*a mathematical description of observed effects*. Like what lenochodek mentioned about the observer effect being a groundless interpretation -- almost everything commonly "known" about QM is complete masturbation, just handwaving about some mathematical relationships that sounds cool. I thought i understood that before, but i dropped out of college before i bothered learning the specific details and it is kind of a shock to learn that these toothpicks of math are what is holding up the entire pop sci edifice of quantum bullshit.

not that i am an expert in QM now, of course, but i am a pretty practiced bullshit recognizer

lenochodek posted:

ty.

toyotathon posted:i think the 3-ball system would be pretty pedagogically boring, so i'm going to skip it, unless a new context arises. i picture it sliding down a track, bumping into walls, and landing in the classic maxwell 2-2-2 coupling. the 4-ball system could be interesting