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Language, concept, number, these we are possessed of and are all part of our grasping of and our relating ourselves to entities. Entities themselves, however, are not in any way possessed of any of language, concept, or number. That we "find" them poetic or majestic, of a kind or a phenotype, or multitudinous or stochastic, although somethings of them, something about their identity, is touchable and accessible by our various abstraction apparatuses, those somethings of them are not themselves linguistic, conceptual, or mathematical. For those, are only us, as they only ever could and should be.

Oops, I got the terms mixed up. Geroch's term was "appropriate," not "adequate." It's his meaning of his term I have in mind: everything in the mathematics has physical meaning and all of the physics one wishes to talk about is describable in terms of the mathematics. Such is an appropriate mathematics for the physics. Some of our mathematics used in physics, I say, hopefully uncontroversially, is clearly a matter of chosen tool, not the mathematical character of the physical reality. Such would be using base 10 in arithmetic calculations and using various coordinate systems. As fruitful as it was to realize that curves can be described by algebraic equations written with reference to a coordinate system, when it comes to geometric facts of curves in the Euclidean plane, which we may take for planes of the physical geometry around us, the method of Euclid we learn in high school for bisecting a line segment is perfect location and physical; no coordinates lain over things by us and used to describe the curves and their intersections add something physical, which we get directly by synthetic geometry (Euclid's way being an example of synthetic geometry, as distinct from analytic geometry).

Given the specific formalization of QM as accepted I would suspect other kinds of numbers as operators or coefficients to the so-called “states” would be improper somehow. I played around with quaternions, more specifically a sort of Mandelbrot generalization to render fractals on a … believe it or not… Amiga computer back in the day. I believe there are multiple imaginary bases i j and k, each squared is -1 but the product of any two is the other (positive or negative depending on the order of multiplication)

I sure did: more on the ways complex analysis is like and different from real analysis (including likeness and difference of R2 and C). I want to watch it again, and I hope to listen also to his part II and part III. The portion starting at about 9:00 was right on our mathematical issue, and I'd expect would be informative to you as it was to me; the part before that is good, but you probably already knew. Are you able to understand each word in that accent? Did you already know it all? (My favorite college mathematics professor was Indian (ordinary differential equations), and so was my favorite philosophy professor, who had done his advanced degree in Göttingen, then was professor in India, then migrated from the Ganges to the Red River in time to get me going on KrV. I love the accent and get every word, due to all that practice, I imagine.) ~~~~~~~~~~~~~~~~ Have you been exposed to the quaternionic formulation of quantum mechanics (Adler 1995)? I do not yet understand the work of Renou et al. well enough to know whether their result also renders any quaternionic formualation of QM short of adequate. By "adequate" I mean as the meaning of Geroch in Mathematical Physics:

Did you learn anything from this video?

Ah yes, this is the convention of using a complex refractive index in calculations to take into account absorption. Quite a convenient use of complex numbers, relating incident light to absorption of light in the material as a function of depth.

The point was not that real numbers cannot be used for relations... that is obvious and you know it. I'll restrain my snippiness to that comment. The point is that in reality you do not literally have "i" quantity of some entity, as such. "i" can be used to help calculate quantities or to represent/characterize (through its odd mathematical qualities) relationships between real things. Do not get me wrong, I am not guilty of reification of real numbers either, they are abstractions we use to directly quantify things in reality, but they are no more real than "i"... merely the directness of their application, and their function differs.

Rand on Discernment of That and What Nathaniel Branden: “Percepts constitute the actual starting-point of human knowledge, in the sense that percepts are man’s first fully aware cognitive contact with the world” (c.1968, 38). The term percept is from Peirce and his contemporaries (see Moore 1961, cited in Rand 1966–67, 2; further, Wilson 2016, 190–95, 204–5). Rand had written in the 1957 exposition of her philosophy: “The task of [man’s] senses is to give him the evidence of existence, but the task of identifying it belongs to his reason, his senses tell him only that something is, but what it is must be learned by his mind.” She defined man’s reason as “the faculty that perceives, identifies and integrates the material provided by his senses.” (Rand was still using that definition in her 1960.) She took human knowledge to run part-and-sum “from the first ray of light you perceive at the start of your life to the widest erudition you might acquire at its end” (1016). “Sensations are . . . an automatic form of knowledge” (1961a, 18). A sensation is “a sensation of something, as distinguished from the nothing of the preceding and succeeding moments” (1966–67). Rand took knowledge broadly enough at times such that sensation, which informs perceivers only that something exists, not what exists, counts as some knowledge. Knowledge for humans would be, in full, “a mental grasp of a fact(s) of reality, reached either by perceptual observation or by a process of reason based on perceptual observation” (1966–67, 45; further, 1970, 84–87). Rand had taken all consciousness fundamentally to be identification (1957, 1016). So all perception, even perception of a first ray of light in infancy, would be an identification. It is therefore not surprising that in her later articulation of Objectivism she would contract her definition of reason to simply: “the faculty that identifies and integrates the material provided by man’s senses” (1961a, 20) in place of “perceives, identifies and integrates . . . .” Rand had it that “sensations are integrated into perceptions automatically by the brain of a man or of an animal” (1961b, 14). Those perceptions in humans are volitionally integrated into conceptual comprehension by reason. Sensations are transitory identifications, not identifying what, only that. Unless a sensation is itself focused upon—say, in neuropsychology—it is not, in Rand’s meaning of the concept sensation, retained in memory, which I cash to mean specifically not retained in working memory or in episodic or semantic memory (i.e., retained only in iconic memory). Conceptualization, conjecture, and inference come under the name reason for Rand by falling under the volitional identification and integration of material from the senses. In Rand’s view, as with Reid and Peirce, the conscious uptake from the senses for the makings of reason is sensory information already automatically integrated into percepts. (See further, Kelley 1986, 31, 44–51, 141–74.) “A percept is a group of sensations automatically retained and integrated by the brain. It is in the form of percepts that man grasps the evidence of his senses and apprehends reality. . . . Percepts, not sensations, are the given, the self-evident” (1966–67, 5). Animals capable of percepts, perceive entities, in Rand’s categoreal sense of that term. Percepts and their objects are susceptible to retention in memory. Peirce had stressed that sense impressions are not first in our knowledge. We are not shut out from the external world, Once Rand had taken on percept and its position in cognition from sensation to reason, I think she really needed to do a little refinement on her 1957 statement that it is only by reason that we discern what an existent is. Animals capable of percepts have some of what a perceived thing is and what actions a thing affords right there. So do we. It remains, of course, that with reason we grasp more, much more, of what a perceived thing is. Additionally, by now it is overwhelming in the neurobiological evidence that into neural activity streams feeding into a percept is a good deal of what a thing is.* None of that formation is volitional, and all of it remains as the given, for conceptualization and reasoning on it. That is, such rich percepts, giving some what in addition to that, can remain first cognitive, aware, contact with the world and sound foundation for knowledge. When we have a percept, it includes places, motions, and some temporal relations in a scene. Are these part of the what a thing is? Or are they only part of the that a thing is? In Rand’s Introduction to Objectivist Epistemology, she articulated some additional metaphysics, and among these additions was the thesis that no existent is without relation to other things. A thing purported to stand in no such relations would be nothing (ITOE 39). That is, there are no concrete existents that do not stand in some external relations. That tunes well with Aristotle: Things “are not such that nothing that pertains to one kind is related to another, but there is some relation” (Metaphysics, 1075a16–17). External relations are there, ready for conscious recognition in percepts and concepts and predications. I suggest that in Rand’s metaphysics and her concept of percepts, her system needs a minor repair by acknowledgement that wheres and whens are within percepts, delivered as aspects of concrete existents, delivered both as that and what of existents *E.g. "Feedforward, Horizontal, and Feedback Processing in the Visual Cortex" by Lamme, Supèr, and Spekreise in Current Opinion in Neurobiology, 1998, 8:529–35. (I'll try to list the References in a later post.)

Real and Complex Analysis (especially over at 9:00 forward)

InfraBeat, Welcome to Objectivism Online! Thank you for the helpful informative post!

Interesting. Newton maintained that numbers as in 7 feet are really ratios, which would be a relation. I've not had time to chase it down, but do you recall an imaginary term that falls out of a classical E-M radiation equation which turns out to correspond to quantity of the radiation absorbed in a medium?

Every once in a while, I'll see an article that sings the praises of the policies of one state or another as setting an example for the rest of the nation to follow. Usually, it's some new welfare-state program or anticapitalist, anti-energy "green" initiative the leftist press would love to ram down everyone else's throats. But this time, we have a whole list from a more or less conservative outlet, Newsweek, which ticks off "The Top 10 Florida Policies Other States Should Emulate." And no, despite how horrible the conservative movement has become, this list includes some ways in which Florida is freer than the rest of the country and doesn't include anything overtly theocratic. It may be true that many states would be relatively more prosperous than they are now if they took this advice, but it is worth considering how they compare to what we would have in a laissez-faire system, where the government is restricted to its proper scope, of protecting individual rights. Here, we'll just list the ten points presented by Newsweek in italics, followed by my brief comments, and my overall evaluation in bold. Regarding these evaluations, I hope to make clear from the context that "good" isn't necessarily full-throated praise, so much as a sense that in today's context, something might be comparatively good or improving about as much as one could hope for given that capitalism is nowhere near the Overton Window. In some cases, my evaluation will differ from whether something actually makes life easier in Florida as compared to other states, such as for uniformity in occupational licensing:Image by Ashley Satanosky, via Unsplash, license. [T]he state has some of the first school choice laws in the nation, ... [and] ... a universal "education savings accounts" bill [is] under consideration in the state legislature... -- The state has no business running education or any other business. There would be no need for such laws in a capitalist society, but this is a step in the right direction. This isn't exactly capitalism, but it is good in today's context. Florida's higher education system is affordable and accessible, from career training through graduate studies. Public college tuition is capped and the lowest in the country. -- Education is not part of the role of a proper government, and I wonder at whose expense tuition is "capped." Furthermore, if there were competition and the possibility of failure, there would be no need for "right to know laws" for paying customers (i.e., students) to have an easy way to judge the value they would be getting from an education. This is par for the course in today's semisfacist mixed economy, and I see no movement in either direction here towards more or less freedom. Neutral. We're one of just a handful of states that lets teenagers get jobs without first needing permission from their school or the state government. -- I had no idea other states were so restrictive. Good. Instead of letting different cities create different and confusing licensing requirements and fees for jobs, our state ensures the rules are the same for everyone. -- All licensing laws are improper barriers to entry to the labor market and should be abolished. Professional associations, independent standards bodies, businesses, and the market should be in charge of determining who is qualified to perform a job. That said, while not having a regulatory patchwork to deal with is easier for people to deal with day-to-day, that masks the harm done by having the state regulate everything. This is bad in a "he makes the trains run on time" sort of way. No income tax. -- Good. Low property taxes. -- There shouldn't be any taxes. This is par for the course today, and since the state doesn't appear to be looking for new excuses to spend our money I'll be generous, call this neutral, and be glad my taxes aren't any higher. Florida recently enacted a law that dramatically speeds up the local approval process for new home construction. -- As with licensing law, the state shouldn't be telling people what to do with their own property, and the "efficiency" here masks impropriety. That said, I'm calling this good in the greater context of so many other states actively discouraging development in so many ways. Florida makes it easy to work from home. -- Regulations that shouldn't be on the books for starting a business have been streamlined. Whoopee. Bad, for the same reason as for streamlined licensing laws. It's one of the few states that requires the legislature to approve the most expensive regulations. -- It will be a long time before the cause of separating the state from the economy gains any steam. In the meantime, at least it's hard to slam everyone with particularly onerous measures. Good. Florida keeps government small... -- Small, but often very improper. This governor in particular has demonstrated the difference between small and proper government on more than one occasion: forbidding cruise lines to test passengers for Covid and forbidding Twitter to ban politicians' accounts come to mind. Absent the governor's authoritarian tendencies, I'd be much happier with Florida's relatively free economy. But it's causing too many people to give the governor a pass on some very questionable policies. Neutral: States should strive for governments of whatever size is necessary to protect individual rights. Any other use of government is wrong.This was an interesting exercise, and one that shows just how bad the state of freedom is in America, which lags badly behind even Florida in most other states. -- CAV Link to Original

and that includes relationships between "portions" of the same thing, so to speak. From what I recall in QM probabilities there is never any "absolute" phase, but relative phase is ubiquitous and the foundation for determining interference, and plays much in what the results of the inner product look like (i.e. probabilities). So real numbers we use to quantifying things but "i" and the like are useful for relating things, in particular phase differences.

No. Real analysis is the study of the real number system, which is a complete ordered field. Complex analysis is the study of the complex number system, which is a field but not an ordered field. (As characterizations of those areas of studies, those are gross simplifications, but they are a start in noting the difference.) Of course, the theorems regarding both are axiomatized by set theory. No, it does not. My remarks in this post are about the mathematics. I'm not commenting in this post on philosophy. I have not studied complex analysis very much, but the most basic definitions and theorems are simple. A very good treatment is found, for example, in Rudin's 'Principles Of Mathematical Analysis', which is one of the classic textbooks. A complex number is, by definition, an ordered pair of real numbers. In other words: Definition: x is a complex number if and only if x is an ordered pair of real numbers. Notation: < > for order pair + for real addition (we'll also define (complex)+ for complex addition) * for real multiplication (we'll also define (complex)* for complex multiplication) - for real subtraction and for the 'negative' symbol Definitions and theorems: df x is a complex number if and only if there exist real numbers a and b such that x = <a b> th <a b> = <c d> if and only if (a=c & b=d) df <a b> (complex)+ <c d> = <a+c b+d> To simplify the notation, we can drop '(complex)' and use just '+' for complex addition, so that both real and complex addition use '+' though officially they are different operations. df <a b> (complex)* <c d> = <(a*c - b*d) (a*d + b*c)> To simplify the notation, we can drop '(complex)' and use just '*' for complex multipication, so that both real and complex mulitipication use '*' though officially they are different operations. df i = <0 1> th i*i = <-1 0> th <a b> = <a 0> + <0 b> th <a b> = <a 0> + (<b 0> * i) (so a is called the 'real part' and b is called the 'imaginary part')

Real numbers characterize and are abstraction we find useful for quantifying things. Imaginary numbers, as directly as I can relate them to things, characterize and are abstractions we find useful for (some specific) relationships between things.

"Analysis" is a complicated formal construct. That which one builds to surround and is supported by real numbers is not the same as that which one builds to surround and is supported by complex numbers. So no, complex analysis is not "really equivalent" to real analysis. These are two different games played in two different arenas... we can and did make them so. Now the idea of analysis is like the idea of a mathematical expression, how it is used or what symbols are there may be different but what it represents or refers to, is not the same as the expression used. The identity operator helps us understand that although the expressions are not the same what they refer to are one in the same, an identity. So I do not discount the possibility that some form of complicated real number based formalism, using Euler relations etc. cannot create something like an answer, i.e. refer to some quantity, which complex analysis also refers to. When used as a coefficient, a real number can be interpreted as a kind of scaling or "quantity of" operator, so taking i, as operated on by 3 you have three of them, so no 3 times i does not remove the i. i however is like a 90 degree rotation operator in the 2d plane we use to arrange our complicated useful contrivances we call complex numbers. EDIT: The last paragraph seems very important to limit and understand the import of their findings.

So mathematically, complex analysis is really equivalent to real analysis? A complex number consists of a real number plus an imaginary number, where the imaginary number has a real coefficient. But using the real number 3 as a coefficient in 3i does not turn 3i into a real number does it?

No one is proposing that, indeed it would be unnecessary. The potential or purported import of the paper is its implication that *which particular kinds* of abstractions we can choose to use in the context, are somehow actually limited.

Sure, we could rewrite the whole thing using only real numbers, but how much additional complication would that create? The more additional complication, the more Occam's razor would say that is not the way to go.

YouTube channel "1420", Daniil Orain, micro-interviews, mid-March 2023, in Moscow. Theme: "Do you think our government can stop this war at all?"

As a non mathematical layperson, it 'feels' like all the maths can describe how much and illustrate a lot of the 'how' precisely by delineating the 'much' , but can not answer any 'why' and especially when 'why' s are not appropriate in a query.

That understanding would be incorrect. Particularly that last sentence. Anything characterizable in complex numbers must be characterizable in real numbers, because every complex number is characterizable in real numbers. As for things, any complex attribute of a thing can be represented by two non-complex attributes associated with the thing. In fact the necessity of characterizing an attribute as (specifically) complex is nothing more and nothing less than having to use more than one (specifically two) real number to characterize that attribute. Every operator and mathematical calculation in the complex plane deals simultaneously with two real quantities, phase and magnitude or alternatively real and imaginary parts. A complex number IS two things, and of a necessity is b0th reducible analytically into those two things and cannot be constituted by less than those two things. It can never be a "simple" number, after all it is a complex number, and it has two absolutely independent components, and cannot be thought of as having any less than two components, and therefore it IS two things. So what is the error? Reification. The fact that an attribute (in a particular framework or theory of reality), to reflect reality has two things associated with it, and that the operators must take into account both, say phase and magnitude, means that a thing has a two-attribute attribute. The reification is an erroneous identification of this two attribute attribute WITH the abstractions we use to work with them... it comes from the way we understand and express and work with this two attribute attribute, namely with complex numbers. Moreover, the simplicity with which we deal with the calculations of the two attribute attribute as if it were one attribute (because they always go together)... every phase must have a magnitude... leads one to believe the formalism as expressed is the only way to express that formalism. That when you change the expressions one has changed the formalism... and that "QM based on real numbers" means somehow trying to use real numbers in a framework built for using complex numbers... The idea of a hypothetical "real quantum state" is nonsensical. Why? Because the referent of the modifier term "real" is purely mathematical or abstract, and the referent of the term "quantum state" is supposed to be an entity of reality. This should be a huge clue to how the authors are thinking...or how they are not being careful about what they are talking about, i.e. what refers to abstractions and what refers to reality. The foundation of QM on the idea of representing reality as states in a vector space, and whose inner product corresponds with probabilities of outcomes. We assign states in the same direction when probability is 1 and assign states as orthogonal vectors when probabilities are 0. We have operators to rotate those vectors modeled on causation and interaction. A nice little game no? It turns out that correspondence between these vectors we have concocted to real world outcomes, requires the use of complex coefficients and operators... but what has that done to the formalism? All that has cone is doubled the degrees of freedom. Sure one could not write QM in the standard formulation, the standard way with real numbers, the correspondence between it and reality requires complex numbers but that does not mean one could not rewrite the entire thing, vector spaces and all using real numbers.

Ron DeSantis, the man who would be "Trump Without the Baggage," faces a dilemma if Donald Trump is indicted in the hush money investigation going on in New York:Trump supporters are pushing for a 'MAGA moat' around Mar-a-Lago to prevent the former president from being arrested as they demand that Florida Gov. Ron DeSantis defy New York authorities and refuse to extradite Trump. ... Now, some MAGA conservatives are demanding that DeSantis refuse to extradite Trump to a 'Soros DA' until the Florida government can conduct its own investigation -- with some even saying the governor should call in the Florida National Guard to protect Trump. [link omitted]The article cites various Trumpist figures who make similar pronouncements, culminating in the following:DeSantis many not have Trump's baggage, but he may trip over it, anyway... (Image by White Field Photo, via Unsplash, license.)Political strategist and former radio host Joey M. also tweeted 'if DeSantis signs that extradition request, his political career as we know it would be over. He'd be worse than Pence. 'If he doesn't sign them, he becomes a hero to MAGA and possibly becomes the de facto nominee for 2024 - or mends fences with Trump and forms a unity ticket with himself as VP.' Meanwhile, [radio host Stew] Peters tweeted 'if Ron DeSantis is the conservative hero he projects himself to be he should REFUSE to honor the arrest warrant for President-in-exile Trump and send the Florida National Guard to Mar-a-Lago to ensure Trump's protection. 'Anything less proves DeSantis is a fraud.'Worse than Pence?! Among other things, apparently including electoral officials who do their jobs even under pressure, Trump's base fails to appreciate that Trump's electoral appeal is ... quite limited ... among other voters. That last point will not be lost on DeSantis, who has managed to pander to Trumpists so far, but now faces the choice of losing the support of this contingent of voters (at least some of whom he'll need to win the Republican nomination) by signing the papers -- or losing the support of non-Trumpist Republicans and independents (whose support any candidate would need against a Democrat) by refusing to sign. It will be instructive to see what DeSantis does and how he justifies it, if push comes to shove. Given that "Trump without the baggage" looks more and more like "a better organized and methodical (read: dangerous to freedom) caudillo" to me, I am not sure that Trump's legal troubles bringing down both men politically would be such a bad thing. -- CAVLink to Original

"For years, it was generally accepted that real quantum theory was experimentally indistinguishable from complex quantum theory. In other words, in quantum theory, complex numbers would only be convenient, but not necessary, to make sense of quantum experiments. Next we prove this conclusion wrong." (12/15/21). In the April 2023 issue of Scientific American, there is an article by the researchers getting the result that the use of complex numbers in standard quantum theory is not really just a convenience, but that some results of standard quantum theory cannot be also obtained in any alternative formulation using only real numbers. In other words, if I'm getting this right, characterization of quantum mechanics using complex numbers (or hypercomplex numbers?), is the only correct mathematical characterization of the physics. That is, purely real-number characterization—however complicated—of quantum mechanics is inadequate as a characterization of the physics.