• What if you dont like the premises of life?
    A lot of people think that by enduring this, that it enriches their life when they make it through. I don't know, for me, it just dulls life that much more that on top of the everyday dealings with other people, BS in general, societal maneuverings of the daily kind, there is the pain and suffering of being struck by enduring illness, injury, and the like.schopenhauer1
    I didn't mean to overcome discomfort through self-improvement, but to suffer through discomfort resiliently (as if stoically, but not really.) In other words, a possible solution is a non-solution.

    As far as romantic love, how does this ameliorate anything? Building a loving relationship, and keeping one, are even more difficult these days than back in the day when it was an expectation (though leading to much unhappiness for staying in bad relationships). Besides, even the best of relationships can lead to pain from differences in expectations.schopenhauer1
    I offered love as motivation, not relief. To remedy the sense of purposelessness, not the sense of helplessness. Love will definitely increase the actual hardship many-fold.

    But anyways, in this more recent climate of shallowness, self-absorption, and short-sightedness, intimate partners are harder to come by these days. The whole caring about someone who is particularly special to you and you to them is diminishing as the years move forward. Increasingly, you're on your own in sickness and health, except for perhaps your immediate family (if they are still alive and well and in communication).schopenhauer1
    I agree. This probably steers into a politically and culturally focused topic, but indeed, people appear to be living isolated. I suspect that there are far reaching consequences - lack of empathy, no sense of responsibility, etc. But I may be over-dramatizing. It certainly doesn't apply to everybody.
  • What if you dont like the premises of life?
    This is the solution. Well, half of it: the other half is learning and teaching. Let both goodness and truth flow into you and out of you, through you, and you will become meaningful to the world and it will become meaningful to you.Pfhorrest
    I can subscribe to this. However, most people, including me, also need a sense of potency to pursue an idea. In the face of life's factors, a person may decide that empathy and enlightenment is futile. I can subscribe to love anyway, because it doesn't really require affirmation of successful outcome to be practiced.
  • What if you dont like the premises of life?
    Nevertheless your response got me to thinking, that this notion of ' spreading the love' in, say, a universal Greek style ( phila, philautia, ludus, agape, etc.) perhaps, might go a long way in achieving that end goal of interconnectedness and purpose.3017amen
    If self-love is experienced properly, as motivation for industriousness, enlightenment, responsibility and self-respect, then I can agree. The problem is, self-love can take any turn. The subject is self-love's own affirmator. Overall however, I do agree that without self-focus and self-awareness, striving to harmonize and assist is futile.
  • What if you dont like the premises of life?
    Well, sure, I care--a little bit, anyway; medication helps. But the cosmos definitely doesn't give a rat's ass that I care. The reason is that the cosmos can't care. The spheres are all silent. They spin. End of their story.Bitter Crank
    I understand. I just wanted to suggest that maybe the problem isn't that the universe is not sufficiently emotive, but that emotion is not what counts towards fixing its colossal issues.

    My personal view: Only the most insensitive, unimaginative dolt would think this is a wonderful world after a careful perusal of life as we know it. Not just for us, but for everything else.Bitter Crank
    Some people just don't concern themselves with life in general, but pursue personal happiness. Some of them are decent people. They just have a different focus in life. That doesn't detract from your statement, which means something different and I agree with it. But I am just saying that not everyone who likes living is the village idiot.
  • What if you dont like the premises of life?
    'All he knows is that he is desperate for love. What he doesn't know is, that he will continue to strive [after he finds love].'3017amen
    Oh. He knows. :) The strife actually increases. But with this kind of strife, there is a sense of purpose. The burden is even greater, but this kind of burden may fill the sense of vacuousness of one's existence.

    We are Beings who are never satisfied; we are trapped in a life of striving (or doing). Much like in our stream of consciousness. After one need is satisfied, it's replaced with another.3017amen
    This kind of consummation approach to love may not be the caring devoted love that I talk about, but I may be wrong. I think that love - romantic or platonic - is supposed to make a person invest effort in someone else's well-being. It is still egotism, but not driven by consummation.
  • What if you dont like the premises of life?
    Because it doesn't make a bad situation worse by figuratively ramming one's head into a virtual brick wall.Bitter Crank
    I know that the OP specifically wanted a non-suicidal option, but rationally speaking, if things are that bad, why not just check out. It doesn't appear a detrimental move from that point of view. Unless the person has a family that they don't want to betray and abandon.
  • What if you dont like the premises of life?
    The cosmos doesn't care.Bitter Crank
    That is interesting. It seems like absurdism, but I wonder what specifically do you mean by the "cosmos doesn't care". That is, you do care, and you are part of the cosmos. So, some part of the cosmos cares. Just not an omniscient, omnipotent, omnipresent part of it.
  • What if you dont like the premises of life?

    I am in that boat as well, so I wonder if the category of human beings under consideration aren't just supposed to accept burden and discontentment as part of life, and deal as anyone in their situation - through resilience and endurance. Then again, this may appear contrary to the spirit of the question, i.e. why have resilience and endurance?

    P.S.: I wonder if loving and being loved isn't supposed to change that attitude. On one hand, love means caring, which implies suffering. On the other hand, caring is investment in life and a sense of purpose.
  • My own (personal) beef with the real numbers
    I think that this is the more accurate representation of what I was saying.Metaphysician Undercover
    I will then answer under the assumption that we are not questioning the efficiency of algebraic numbers as they are used in the design and analysis of numerical algorithms for daily and industrial use.

    I don't agree with your conclusion though, because it requires that we make a complete separation between physics and mathematics. Suppose we assume such a separation. Mathematicians just dream up their axioms and principles for no apparent reasons, just because they are beautiful or something, so that the mathematical principles are somewhat arbitrary in this way, pure mind art.Metaphysician Undercover
    I am not saying that it necessarily has to be pure mind art, but it wouldn't change the assessment we make of the final product. Even if our improvements in the scientific method (the philosophy of mathematics) remove obstacles to its continuous development, it should not come at the cost of the numerical properties of the prescribed computations. Just as long as the scientific method produces a framework of computations and analysis for our varied conventional applications that conserves the known accuracy-complexity tradeoff, the improvement can venture in any desired direction. But exploring concepts and algorithms of computations whose conventional (daily and industrial) utility is inferior to the ones currently in use is not an option, no matter what methodological grounds we have for that.

    They leave these principles lying out there, and the physicists pick and choose which ones they want to use. It's like a smorgasbord of tools lying on the table, which the physicists can choose from. That's a fine start, but we must respect the fact that the process is reciprocal. So once the physicists choose their preferred tools, and start using them, then these are the principles that the mathematicians are going to concentrate on improving and fine tuning.Metaphysician Undercover
    This is true. But mathematics is not focused only on fundamental physics. It serves economics, construction engineering, software design and electronics, government, etc. All those fields are served on equal footing by mathematics. Physics is undoubtedly the most fundamental of sciences. But that does not mean that mathematics is physics oriented. It serves any situation involving computing and conceptualizes theories that unify as many applications as feasible. If a given field requires specific axiomatic framework, it needs to engage the mathematical community deliberately and the other fields do not have to be impacted.

    Evidence of this is the fact that the real numbers came from the use of rational numbers, and the use of geometrical principles which created irrational numbers.Metaphysician Undercover
    Yes, but this happened in times when people believed the earth was flat. How irrational numbers were originally conceived is irrelevant to their contemporary meaning.

    Mathematicians did not have to allow irrationals into their numerical principles, but they were being used, so the mathematicians felt compelled to incorporate them .Metaphysician Undercover
    The way I interpret irrational numbers today is as a property of the process you use to compute approximate quantity when a diagonal, perimeter, the intersections, etc, of various objects of various shapes are involved. The question is not what best explains the underlying physics, but how can we compute economically and socially interesting factors - the amount of material necessary to manufacture a part in the desired shape, the heat transfer occurring at the surface of a container, the sliding resistance a material with a given shape has, etc. We are not investigating the underlying physics, or trying to achieve precision in excess of what we need, but merely seek an efficient method for computing values within the desired accuracy. Irrational numbers are our "design" of this method, not our "interpretation" of the objects involved. We can still investigate the physical fundamentals, but this is not a concern for our procedures.
  • My own (personal) beef with the real numbers
    The physicist applying mathematical principles believes that the "materials" of mathematical ideals are a good approximation of the "materials" involved in actual physical structures, when in reality they are not.Metaphysician Undercover
    Maybe I am not on the same page with you here. Before I try to answer in any detail, I have to get clear on the substance of our conversation. Are you saying that you imagine a system of computations which does not involve algebraic numbers, and either has the same utility for our industrial and daily applications at a discount computational cost, or enhances our conventional applications at no additional computational cost? Or do you mean that the use of irrational numbers is conceptually inaccurate with respect to a first-principles analysis of physics? If the latter, as I said, I don't think that it matters for mathematics.
  • My own (personal) beef with the real numbers
    Now we have entered into an extremely confused and contradictory conception within which distinct things are said to be distinct particulars, and they are treated by the application of the theory as distinct particulars, yet they are stipulated by the assumptions of that same theory to be the same in an absolute way. That's the kind of mess which "grain uniformity" might give us.Metaphysician Undercover
    I think that you are ascribing to mathematics the kind of role that I don't think it has. At least, not directly. Or maybe I did walk into this when elaborating over my example. Its aim wasn't to model the structure of physical objects, but to illustrate how coarse structures not literally represented by mathematical ideals, can still be usefully approximated by those ideals. It was designed to have some similarity with the atomic structure of materials. But it is not a theoretical model for physics. The idea was, that actual physical structures approximate the mathematical ideals, and our numerical algorithms approximate those same ideals, and thus, under certain assumptions of the magnitudes of the involved deviations, our numerical algorithms match the physical structures within the required precision.

    About the grain uniformity. let's say that we are talking about Voronoi partitioning of space, and not triangular tessellation, because it simplifies the definition. When I say uniform grain, what I mean is that the distances between the generator points of the partitioning follow some known distribution with finite variance. This is a simple example, and not a physical model. However, it does illustrate, I think, that the diagonal will usefully approximate some quantity over a rough spacial structure, despite being itself defined as a continuous object.
    But approximation in practise is not the same as approximation in theory.Metaphysician Undercover
    Which theory do you mean? For me, real numbers theorize some characteristics of computation. And do so imperfectly. The algebraic structure is defined over some converging computational sequences. It does allow for imaginary objects that do not correspond to actual computational processes, because the latter can not be specified procedurally. And thus it works with incomplete specification. But it is a best effort theory. I am not necessarily subscribed to the idea that real numbers correspond to the points of physical lines, whatever that may mean. It may turn out that this conceptualization works, but I am not convinced. I do think that it works for approximations however.

    P.S. I want to clarify that I do actually think that our computational and logical ideals are naturally inspired. They are not literally representative of any particular physical structure, but they are "seeded" as concepts by nature, whether our sentience existed or not.
  • My own (personal) beef with the real numbers
    The problem I've been discussing is that whatever it is which is expressed as "a square" does not actually exist in "concept-space" because the perpendicular sides are incommensurable. There is a deficiency in the concept which makes it impossible that there is a diagonal line between the two opposing corners, when there is supposed to be according to theory. The figure is impossible, just like the irrational nature of pi tells us that a circle is impossible.Metaphysician Undercover
    If you view mathematics as explanative device for natural phenomena, I can certainly understand your concern. However, I see mathematics first and foremost as an approximate number crunching and inference theory. I do not see it as a first-principles theory of the space-time continuum or the world in general. I see physics and natural sciences as taking on that burden and having to decide when and what part of mathematics to promote to that role. If necessary, physics can motivate new axiomatic systems. But whether Euclidean geometry remains in daily use will not depend on how accurately it integrates with a physical first-principles theory. Unless the accuracy of the improved model of space is necessary for our daily operations or has remarkable computational or measurement complexity tradeoff, it will impact only scientific computing and pedagogy. Which, as I said, isn't the primary function of mathematics in my opinion. Mathematics to me is the study of data processing applications, not the study of nature's internal dialogue. The latter is reserved for physics, through the use of appropriate parts of mathematics.

    The problem in this situation is that the indeterminacy is created by the deficient theory. It is not some sort of indeterminacy which is inherent in the natural world, it is an indeterminacy created by the theory. Because this indeterminacy exists within the theory, it may appear in application of theory, creating the illusion of indeterminacy in the natural thing which the theory is being applied to, in modeling that natural thing, when in reality the indeterminacy is artificial, created by the deficient theory.

    We can allow the indeterminacy to remain, if this form of "concept-space" is the only possible form. But if our goal truly is knowledge, then it cannot be "the most fitting solution" to our problems.
    Metaphysician Undercover
    As I said above, I don't think that mathematics should engage directly to enhance our knowledge of the physical world, but rather to improve our efficiency in dealing with computational tasks. It certainly is a very important cornerstone of natural philosophy and natural sciences, but it is ruled by applications, not natural fundamentalism. At least in my view.

    I'm not sure I actually understood your example. Maybe we can say that Euclidian geometry came into existence because it worked for the practises employed at the time. People were creating right angles, surveying plots of land with parallel lines derived from the right angles, and laying foundations for buildings, etc.. The right angle was created from practise, it was practical, just like the circle.Metaphysician Undercover
    This is what I mean.

    Then theorists like Pythagoras demonstrated the problems of indeterminacy involved with that practise.

    Since the figures maintained their practicality despite their theoretical instability, use of them continued. However, as the practise of applying the theory expanded, first toward the furthest reaches of the solar system, galaxy, and universe, and now toward the tiniest "grains" of space, the indeterminacy became a factor, and so methods for dealing with the indeterminacy also had to be expanded.
    Metaphysician Undercover
    I do not see how our newfound knowledge about the universe will impact all of the old applications. How does it apply to the geometries employed in a toy factory, for example. The same computations can be applied in the same way. Unless there is benefit to switching to a new model, in which case both models will remain in active use. This is the same situation as using Newtonian physics instead of special or general relativity for daily applications. It is simpler, it works for relative velocities in most cases, and has been tested in many conventional applications. I am breaking my own rule and trespassing into natural sciences, but the point is that a computational construct can remain operational long after it has been proved fundamentally inaccurate. And therefore, its concepts remain viable object of mathematical study.

    Now, to revisit your example, why do you assume "grain uniformity"? Spatial existence, as evident to us through our sense experience consists of objects of many different shapes and sizes. Wouldn't "grain uniformity" seriously limit the possibility for differing forms of objects, in a way inconsistent with what we observe?Metaphysician Undercover
    Of course it would. I meant applications where the grain is indeed uniform, such as the atomic structure of certain materials. And even then, only certain materials would apply. The point being is - every construct which can be usefully applied as computational device in practice deserves to be studies by mathematics. As long as it offers the desired complexity-accuracy tradeoff.

    I do agree that the use of mathematics in real applications is frequently naive. And that further analysis of its approximation power for specific use cases is necessary. In particular, we need more rigorous treatment that explains how accuracy of approximation is affected by discrepancies between the idealized assumptions of the theory and the underlying real world conditions. I have been interested in the existence of such theories myself, but it appears that this kind of analysis is mostly relegated to engineering instincts. Even if so - if mathematics already works in practice for some applications, and the mathematical ideals currently in use can be computed efficiently, this is sufficient argument to continue their investigation. Such is the case of square root of 2. Whether this is a physical phenomena or not, anything more accurate will probably require more accurate/more exhaustive measurements, or more processing. Thus its use will remain justified. And whether incommensurability can exist for physical objects at any scale, I consider topic for natural sciences.
  • My own (personal) beef with the real numbers
    Sorry simeonz, but your example seems to be lost on me. The question was whether whatever it is which is represented by √2 can be properly called "an object". You seem to have turned this around to show how there can be an object which represents √2, but that's not the question. The question is whether √2 represents an object.Metaphysician Undercover
    Probably I don't understand the point of the conversation. But just to be clear. The space tessellation/partitioning was not to show how one mathematical construction can be derived from another. The tessellation corresponded to some unspecified physical roughness with uniformly spaced, but irregularly situated constituents. It aimed to illustrate that solving world-space problems imperfectly (due to efficiency constraints) results in the adoption of a modus operandi solution, whose own structure exists only in concept-space.

    For analogy, similar utilitarian interpretations exist for probability and statistics. Most classical (pre-QM) applications of probability are not related to genuine physical indeterminacy, but to making decisions with imperfect knowledge of the conditions. The choices are made according to some sense of overall utility, independent of the true and objectively predictable, but unknown individual outcomes (Pignistic probability). In other words, we conceptualized indeterminacy, not because of its objective existence (aleatoric uncertainty), but due to our lack of specific knowledge in many circumstances and because introducing indeterminacy as a model was the most fitting solution to our problems.

    I believe that all mathematics have utilitarian sense to them. Square root 2 is object of thought and human decision making, not of some independent world-state. Its existence may have physical grounds, but doesn't have to be perfectly accurate. It is put forward to deal with solving problems through computation. At least this is my way of thinking.

    As I said, I may misunderstand the topic of the discussion altogether, which is fine. But just wanted to be sure that the intention of my example was clear. (That is - that the space tessellation was not intended as a mathematical structure, but as representation of some unknown coarseness of the physical structure, being ignored for efficiency reasons.)
  • My own (personal) beef with the real numbers
    But if we define "symbol" in this way, then we cannot use algebra or set theory, which require that a symbol represents something.Metaphysician Undercover
    I am trying to honestly understand, but why do you propose that sets should only include apriori existing entities, and not ones defined by the processes of inference and computation themselves. That is - logic is an algorithm and our application of that algorithm manifests the imperatives in the axiomatic system. The algorithm is inaccurate in almost all practical cases, and therefore is not exactly representative of apriori existing objects.

    The real question though is whether the "object" supposedly represented by √2 is a valid object. If you assume as a premise, that every symbol represents an object, then of course it is. But then that premise must be demonstrated as sound.Metaphysician Undercover
    Let me try. Suppose that people have to compute the ratio between the lengths of the sides and the diagonal of an object that approximates a square, but lives in some unknown tessellation of space. You are not informed of the structure of the tessellation apriori and you know that the effort for its complete description before computation is prohibitive. You do however understand that the tessellation is vaguely uniform in size and has no preferred "orientation" or repeating patterns. It is random in some sense, except for the grain uniformity. The effective lengths for the purpose of the computation are the number of cells/regions that the respective segment divides. You also know that the length will be required within precision coarser then the grain of the tessellation/partitioning itself. With this information in mind, you want practical algorithm for the computation of the ratio between the sides and diagonal of a square, to an unknown precision, which is greater then the grain of the tessellation.

    Now, honestly speaking, I cannot prove that the most effective generalized answer should be square root 2. But this is the kind of problem that an engineer would face. And if not theoretically, they could confirm empirically if square root 2 is a good general approximation. What would you advise? And when analyzing your algorithm, what object would you introduce to compare its convergence to other algorithms?
  • My own (personal) beef with the real numbers
    I think that people develop intuitions, depending on whether they consider mathematical objects as rigorous philosophical metaphors for some physical counterparts. As I said, when I think about mathematics, I reason from an information processing perspective, where knowledge itself is the object of investigation. For me, the diagonal of a square is "square root nothing", because perfect squares rarely exist in our daily experience. At the atomic and subatomic level, perfect geometries may actually make sense (at least probabilistically), but that doesn't explain how we managed to exploit mathematical ideals for so long. I think that the information revolution pushes the emphasis gradually from natural sciences, towards a more "means to an end" perspective of logical studies.

    For me, when we construct mathematical models, we are factoring in uncertainty, efficiency, capacity (for control, measurement, computation, etc), and we concoct a useful approach to solving a practical problem. This does not mean that our solution is completely disconnected from the underlying reality, but it is not literal representation of the natural phenomena either. It is mostly practical and has many interesting properties on its own. Those properties determine how it behaves when we use it and even if they are not necessarily properties of the original system (although they could be), they are worth investigating if we plan on applying the solution many times and want to have understanding of its behavior.

    So, to me, algebraic structures, real numbers in particular, are "ways" of dealing with problems. They are inspired by nature, but are not necessarily literally representative of natural objects. The specifications of those algebraic structures, the solutions based on those specifications, etc, have interesting, in some cases paradoxical properties, that are worth investigating simply because of our usage of those structures. If they happen to coincide with nature's geometries on some fine-grained level, this is particularly interesting, and bodes further investigation, but it doesn't affect the originally intended utility.
  • My own (personal) beef with the real numbers
    I agree with you on the square root of 2, of course! But I am not so convinced that mathematical objects are only cooked-up fictions not related to physical reality.Mephist
    I will allow myself to interject, although the physics involved in your discussion appears beyond my competence. In any case, just because something is not physical, doesn't make it purely fictitious.

    For a lame example, if I define a geographical location called "not in London", which has the property that any statement exclusive to the London area is untrue for that abstract location, and all other statements remain undefined, this area would not properly represent any part of the universe. Why? Because many statements can be made about the Universe that are not specific to any location, and remain true for "not in London" in practice, but are not included in my structure/axiomatic system. However, I am constructing "not in London" not to express specific knowledge, but to express my lack of specific knowledge. I am creating an abstract entity which expresses my epistemic stance.

    This is the way I look at mathematical objects in general, and real numbers in particular. They can be physically represented, if they happen to be. But generally, they are specifications more so then anything. As all specifications, they express our epistemic stance towards some object, not the properties of the object per se. Real numbers signify a process that we know how to continue indefinitely, and which we understand converges in the Cauchy sense. Does the limit exist (physically)? Maybe. But even if it doesn't, it still can be reasoned about conceptually.
  • My own (personal) beef with the real numbers
    IMV, the rationals are quite difficult. We know how students hate fractions. But I like the idea of 1/n as a kind of flexible unit. Then m/n is just m of those units. We can adjust n to increase or decrease resolution. And we can do various conversions. So it's difficult but still (after much work and thinking) ultimately intuitive. At least for me.mask
    I understand. After all, this is the rationals' whole gimmick - they are dense. Of course, the finite decimal fractions rationals I talked about earlier are also dense, and easier to compute. But they are not a field. So starting with integers, you cannot scale yourself arbitrarily.
  • My own (personal) beef with the real numbers
    One can construct the positive real numbers as a simpler version of the cuts. In the version I like we have a ray of rational numbers that starts from zero (a subset of Q that is closed downward with no maximum.) I like this for its intuitive connection to magnitude/length.mask
    This is exactly the geometric interpretation that got me into trouble. :) It assumes that rays have points corresponding to every non-negative real number (or lines have points corresponding to all real numbers.) To which, I remember my brain screamed, how do you know? Of course, if this is just analytic geometry, it would be true by construction, but then the argument becomes circular. So, I was asking essentially, how do we know that lines/rays, as they appear in real life, are complete. They could be, or they might not be, but how would a mathematical textbook use something like that, that we know very little about (i.e. space), and which is not axiomatic in nature, and use it to define a mathematical concept. At least for me, it didn't work, and caused me some difficulty.

    I suppose this might be related to the objection that the OP has.

    P.S. What I mean is - the definition (edit: interpretation) works if the person reading it has the right attitude. But I am rejecting its use on methodical grounds, of being informal. (And untested.)

    P.S.(2): The reason why I got so heavily stuck on this interpretation was not so much because it prevented me from making sense of the mathematics presented, but because I started to question what was the goal - were numbers lexical entities or geometric properties? What was that we were trying to define - quantities, computations, geometrical facts? How could we validate them? At this point I didn't know much about algebraic structures and axiomatic systems. I went completely on a pseudo-philosophical tangent and refused to learn anything whose methodological grounds I did not understand completely.
  • My own (personal) beef with the real numbers
    the reals can be built as cuts or Cauchy sequences. I like cuts for not being equivalence classes. It's an aesthetic preference. Cuts are beautiful ('liquid crystal ladders').mask
    Actually, yes. Dedekind cuts are another constructive approach. Not too different in spirit, I would say.

    Was it Cantor who said the rational numbers are like the stars in the night sky and the irrationals are like the darkness in the background? Perhaps this has been posted before.jgill
    Rational numbers are actually quite nasty, if you want to work with them in computations. They are pleasant, if you are performing a finite number of arithmetic operations. But assuming a "fraction" representation, once you start evaluating some recursive formula, the numerator and denominator become unmanageable quickly. I am not sure how fast the periodic part in the repeating decimal representation grows, but I wouldn't want to work with that either. That is why software uses only fractions with denominator in exponent form (*) and represents the rest approximately. Correspondingly, algebraic numbers, and even computable transcendentals, are not that bad, when compared to arbitrary fractions.

    I am only stating this, because rationals are looked upon so favorably, but I find that their simplicity is somewhat overstated.

    * finite digits after the decimal point in appropriately chosen base;
  • My own (personal) beef with the real numbers
    I think that we might come from different understanding of what the real numbers "should" represent in modern mathematics. My real analysis textbook introduced real numbers as corresponding to points on the "real line". I was confused by this explanation and got stuck on researching fundamentals in other textbooks. Almost failed the class due to mismanagement of my time. In retrospect, the definition was characteristic of the old-school soviet style of math textbooks. It assumed that all mathematical objects should be considered metaphors for physical systems. This is not the contemporary view, in my opinion. And it is not my view anymore. For me, mathematical objects are pure concepts.

    For real numbers, I consider two interpretations to be pedagogically correct. The first - algebra over totally ordered equivalence classes of Cauchy sequences/processes. This is the concrete/applied way to interpreting them. The fact that those equivalence classes have order and algebra defined over them does not imply that they stand on equal footing with the approximating elements of the sequences themselves. But nonetheless, they do act amorphously enough to allow us to think of them as "quantities". We call them numbers, but we also call complex numbers such, and they are not even totally ordered. So, to some extent, it is just a matter of nomenclature.

    We deal with algebras on equivalence classes of Cauchy sequences, because we are interested in the convergence properties of the sequences, as well as how convergence interacts with transformations of various kinds - i.e. whether functions are continuous or not, whether operators are defined when acting on maps for those classes, etc. But ultimately, it still boils to our interest in the concept of convergence, not the reals for the sake of the reals. The real number in this sense is just a specification of the approximation process, whose behavior we need to analyze. Specifications can be, but need not be physically represented.

    Alternatively, the notion of a real number from abstract algebra is one of a complete ordered field. Ultimately, it is the same concept. The properties are the same, except that the approach to the investigation is leaning more heavily towards non-constructivism. Which is fine, because this is what abstract algebra is all about. In fact, in some sense, the abstract definition is the proper definition, and the constructive one serves as an illustration. The latter is pedagogically necessary, but once understood, is not essential anymore.

    As I said before, I cannot see how the study of complete ordered fields, being equivalent (up to isomorphism) to the ordered algebra of equivalence classes of Cauchy sequences can be replaced with something else, without reducing the scope of the theory. You either have analysis of your objects, or ad-hoc usage, but the latter is just a trial and error.
  • My own (personal) beef with the real numbers
    The way I reason about it (ie, as a software engineer), real numbers specify the convergence characteristic of approximation processes that deal with real world problems. What you are saying is that people should study the numerical methods that approximate real world solutions, but shouldn't study analysis of this essential characteristic, which seems to me questionable. Maybe your point relates to the general debate in society - whether engineers should study only constructions and hands-on skills and not analysis (how to derive properties of those constructions), but even then I am leaning towards the usefulness of theoretical understanding.

    In Eastern Europe, software engineers studying for bachelor's degree have real analysis, abstract algebra, differential equations, etc, as mandatory subjects. That much is indeed true and many are dissatisfied with the curriculum for being too math heavy.
  • Why do you think the USA is going into war with Iran?
    I am not an American, but the population always disapproves government decisions that were not properly communicated. No matter who's right or wrong, surprising the voter is never a good policy. Nothing should transpire without popular consent or at least discussion and feedback of some kind. Obviously, I don't mean the particular strike, but the need for escalation, if and when necessary. Another problem here is that Trump doesn't always stick to his word, and while he may have spoken about the possibility of military actions many times, his remarks would not have been considered credible at the time.
  • The Notion of Subject/Object

    Let's say, I need to drink water from the waterhole, I like eating grass, and I am afraid of meeting lions. Who cares if I have established my identity of an antelope? I may not acknowledge that I am me (which obviously strips me of some advanced reflective mental faculties), but my agency still creates a subject's point of reference. My thought process needs to be coherent in order to induce a subject, because we mean something sustained by this concept. If I liked eating grass, but spontaneously decided to mate with a tiger I saw at the waterhole (I hate myself for this example), then my identity would be too distorted to induce a meaningful subject's point of reference. There would be no subject per se, just a collection of random ideas occurring without any rhyme and reason.

    In a sense, I see the subject as the property of a collection of thoughts, of being organized to manifest some consistent agency.
  • The Notion of Subject/Object
    Thoughts, strictly speaking, are the one thing that does NOT have subject/object dualism proper.Mww
    I did not mean to say that all thoughts refer to the subject. In fact, I did not address (or honestly even think of) the distinctions between awareness, knowledge, self-awareness, etc. Indeed, only self-aware thinking incorporates the subject explicitly. Still, self-interest is present in most thought processes - even animal ones. So, although it is not formally present, the subject still emerges "organically", so to speak, from the coherent pursuit of personal advantage. Even if it is not directly expressed by the said thoughts.

    Internally, the thinker is the thought.Mww
    Actually, this is exactly what I meant. That the subject arises in consequence from the coherent pursuit of self-centered objectives by the individual. This does not even imply self-awareness, unless the subject becomes the object of discussion itself.
  • The Notion of Subject/Object
    I’d consider them machines.Xtrix
    You probably mean objects incapable of being subjects. They will be machines by definition, no matter what. Or do you mean, that we are not machines, or fundamentally distinct from machines? If so, how?

    Maybe in the future they’ll acquire “consciousness” of some kindXtrix
    It is not obvious to me what does it mean for something to "acquire" consciousness. Is this a behavior modification or substance change or some other metaphysical phenomenon? Because stated in this way, how does one challenge any claim that something has or hasn't acquired consciousness. Also, it isn't clear to me what consciousness denotes - a behavioral pattern, a type of experience, etc. If it is a type of experience, how can a person know that it exists outside of their own being - i.e. the solipsism style argument.

    but we’re a long way out from that.Xtrix
    Is time relevant? Or do you mean that the emergence of such advanced AGI is suspect to you for some fundamental reason?
  • The Notion of Subject/Object
    But not all objects are subjects it would seem, unless you attribute to rocks conscious awareness, which I doubt anyone would.Xtrix
    What about AGI in computers. Hypothetically. We can either presuppose that despite any behavioral characteristics machines are to rocks and planets different from what human beings are to them. Or otherwise, we can compare ourselves to rocks in our role of subjects by comparing them to machines first. Do we consider rocks as self-aware as ourselves? By many degrees of magnitude they lack the expressive, cognitive and reasoning parameters for that. But is the quality binary or is their sentience so insubstantial that it borders inconsequential by our usual standards. If the quality is binary, which side of the filter would AGI computers fall? And how different are computers from rocks then, fundamentally? A binary distinction I think already assumes an exosystemic component - i.e. a mind-body distinction.

    P.S.: Maybe only thoughts have subject and object proper. And the human being is a vessel that serves as a locus for a collection of coherent thought processes that give rise to a sense of identity. These "thoughts" interact coherently to form the role of a "subject".
  • Christianity and Socialism
    Semi-serious. I mean, if you account for all the years the Catholic Church has been around, and combine their liquid and non-liquid assets, along with probably being the first to invest in the stock markets, you should end up with a hefty sum of money accumulated throughout the years, no?Wallows
    Yes, but money does not necessarily equate prosperity. I could make the argument that the unshaken faith of its followers is a much more important asset for any church.
  • Christianity and Socialism
    Yeah, it's definitely a message in the New Testament that often gets left out.Wallows
    My bad. I thought you were serious there.
  • Christianity and Socialism
    Yes, it does make one wonder just how the Vatican or Opus Dei maintain and expand their wealth through the stock market.Wallows
    Sorry, but honestly, I am not informed on the subject.
  • Christianity and Socialism
    I see that we're talking about money, so I might as well comment that the only instance where I picked out the fundamentally humanistic trait of Jesus, was his encounter with lenders/shylocks/money-lenders in the Bible. I don't think there's another instance in the New Testament where Jesus is infuriated more-so than towards money-lenders.Wallows
    I think that most people will agree that banking and stock trading (especially day-trading), even though being legitimate ways of making money, leave a lot to be desired in terms of ethical underpinnings.
  • Christianity and Socialism
    I don't have a problem with most of what you said. I would like to point out that automation has caused many jobs to disappear. Believe it or not automation has even taken away software development jobs. Developing software 50 years ago believe it or not was more contingent on an understanding of discrete mathematics where as in this modern age it is surprisingly much more competitive and relies more on memorizing APIs.christian2017
    This problem is actually going to get much worse, and I believe may cause the dissolution of societies as we know them today. It is frequently ignored, but it is the elephant in the room, and will have to be addressed sooner or later.

    As far a globalism goes, i would rather be tempted to buy a $200 dollar toaster made in my own country then a $8 dollar toaster made in china, given the fact that I would more likely be paid a living wage if i worked in a factory.

    I believe people in America would be more happy with a better job and at the same time having less material possessions due to the cost of labor.
    I understand your sentiment, and as I said, the use of cheap labor or the export of industries is abusive, because it transfers the economical welfare accumulated through the people of one country to a different location, with the difference becoming personal wealth of the entrepreneur. Even though, personally speaking, my country can use the investments, I cannot deny that I see a "glitch" in that.
  • Christianity and Socialism
    I agree with that mostly except that money can be manipulated much easier then land, resources and services. However it is even possible to manipulate the relationships between people regarding land, resources and services.christian2017
    The problem with using commodities and natural resources directly for barter is that they have limited application. Money have unlimited application, abstractly, hence transacting with them is much more powerful.

    I agree, but during the middle ages they did have monasteries which many poor families sent their children too.christian2017
    Even if the politics in this regard were standard, I suspect that a lot of the wealth of the church was accumulated through state funding, land ownership, or donations from the wealthy aristocracy. But this wealth came at the expense of the poor, whose rights were trumped in favor of their lords. Therefore, the pity offered in this way was not an entirely positive effect.

    Many of the problems we have today are a distant extension of the industrial revolution. Automation, Globalism and money manipulation have made it hard for many poor people to be self sufficient.christian2017
    The industrial revolution was even worse than the middle ages. And that says something. It is one of the grimmest periods in human history. When someone talks about the success of western capitalism, I always think about the initial price that was paid - slavery in south US and children working to death in Great Britain. Nonetheless, times have changed for the better.

    Regarding money manipulation, as I already said - this is abuse of an instrument. This is not an excuse for the misfortune it causes, but the balance will be judged differently depending on the person's situation. If you take a non-electural government scheme for central welfare distribution, the same issue arises, because you have to rely on correctly functioning meritocratic system of appointments to office, and if it fails, you have a different kind of monster.

    Regarding globalism, I am not sure what you mean. Different people have different issues with it. Do you mean the introduction of cheap labor into countries with high economic standards, cultural infusions, price pressure from imports, etc. To be honest, I do think that some of those effects are indeed abusive in a very specific technical sense (which I don't want to elaborate right now). At the same time, in any competitive situation, the person who is willing to sacrifice the most defines the expected performance - there is no level playing field. This turns any competition into terror experience for the participants. But unfortunately, I believe that natural competition is required for unbiased evaluation of performance - anything else is a test of some kind of norm or preference, which is not an objective test.

    The suicide and opiod abuse rate in the US is extremely high.christian2017
    I cannot comment on that. Maybe the capitalism in the US is managed poorly compared to other countries indeed. Yet, I don't think that I have ever seen a statement that capitalistic countries have higher suicide and substance abuse factors in general.

    I would argue many modern Americans have become very fierce in their outlook on life due to the fact that in some sense American devalue human life more than any other people in the past 2000 years. I believe the Medieval man very often acted as a coward because they enjoyed life more than we do.christian2017
    Maybe, or maybe they didn't know any better. Notice the rebellion I outlined in my second reply. It hasn't ended well for the poor folk.
  • Christianity and Socialism
    Here's what I surmise. In countries where interpretations of the Bible are guided by an authority (Catholicism, Eastern Orthodoxy), we have seen the emergence of socialist tendencies and doctrines thereof. Think, Latin America or even Eastern Europe, with the USSR being a quizzical example.Wallows
    Actually, being from East Europe, I can assure you that the connection is mostly tangential. The more to the east a culture is, the more preference for authority you find in it. The more to the west and north you go, the more egalitarian and democratic cultures you find. In East Europe and Asia, people mostly distrust the individual, and they prefer external government. The USSR communism was not partial towards religion, but it afforded it some existence. Most people disassociated with their religious beliefs at the time, or would not discuss them openly in society. Interestingly though, a lot of ex-party members today are fervently religious. I think that they want to have external authority that dictates normative behavior, and as communism exited the political scene, they found religion a suitable replacement.

    Altogether, I do not think that half of the people describing themselves as theists in East Europe are actually philosophically religious. This is more likely just a customary state for them.
  • Christianity and Socialism
    I think if people understood that money is a legal fiction (Noah Harrari) they would be less angry over the issue of welfare.christian2017
    Money are symbolic for insured debt. I wouldn't call "debt" a fiction anymore then I would call "promise" a fiction. The problem with money is that by design, it is intended to be accumulable. This ensures that individuals who produce in excess of their needs, by virtue of their savings, can recall their debts from society, and thus enterprise a locus of collective human effort. Unfortunately, this also gives them power over the collective that they thus manage, which enables them to extract disproportionate debt from society in some cases. Eventually, this can result in monopoly, plutocracy, etc.

    Nonetheless, with all its deficiencies, I would call money an abstract agreement.
  • Christianity and Socialism
    Maybe to guide the conversation, I would propose the edifying question as to why Marx was bashing on Christianity or any organized religion to such a degree to declare it the opiate of the masses? The divorce between socialism and Christianity seems to have been declared at this point in history.Wallows
    As is usual, I am reaching the limits of my narrow philosophical competence. But I propose that the conversation does not equate Marx to modern socialism. In fact, it should be modern communism, because western socialism appears to me to have divorced itself from the more radical Marxist ideas. I don't think that all modern conservatives are atheists theists as well. I am unsure where the divide currently stands - taxation politics, state interference in the market, individualistic vs collectivist ethics.

    Regarding Marx - from what I know - his attitude was humanist, which I understand to mean that the accountability of a person is first and foremost to his society. That the people create the ethical standards that guide the individual, not religion. And he probably did not believe in the ontology of the religious teachings as well - miracles, deity, etc. Where humanism and western Christianity differ in their outcome will depend on interpretations. I can certainly see compatibility between them, if the person has the right attitude.
  • Christianity and Socialism
    I see you're going further back than I had anticipated. Please elaborate on this fact of the aristocracy adopting Christianity and thus making it politically conservative?Wallows
    One could begin with the acceptance of the divine entitlement to privilege, indirectly stemming from the king's own divine right. This notion was used as justification for suppressions of civil insurrections. In the same article you can find the following account:
    One passage in scripture supporting the idea of divine right of kings was used by Martin Luther, when urging the secular authorities to crush the Peasant Rebellion of 1525 in Germany in his Against the Murderous, Thieving Hordes of Peasants, basing his argument on St. Paul's Epistle to the Romans 13:1–7 — Wikipedia
    There was a merger between the political/legal and religious systems, where the latter suppressed reform from within (in spirit) and the former suppressed it from without (by force.)
  • Christianity and Socialism
    I read somewhere that the popularization of Christianity was helped by the public disaffection for the elitist ethics in the Greco-Roman polytheistic religions. The latter celebrated exceptional merit, exceptional heroism, exceptional strength, exceptional ancestry, which would not be perceived as relatable to the weakened and fearful enslaved and plebeian masses. Furthermore, the mythos of the ancient world was hard, punitive, and unforgiving. Christianity may have been partly embraced as a source of self-confidence for the people, affording space for their personal weaknesses and unequal social standing.

    I also find it hard to accept the conservative argument, that revolutionary change should be avoided when possible, because of its destabilizing consequences, when the very religion around which they center their own narrative was among the most revolutionary cultural changes of its time and its region. This changed during the Middle ages when the aristocracy made religion their own prerogative again. (Probably this is also the period when Christianity became politically conservative.)

    I am neither particularly left-leaning politically (in spirit maybe, but not as a political system), neither conventionally religious, but I am interested by this argument.
  • The bijection problem the natural numbers and the even numbers
    Informally, and for so long as you and your readers understand the formality underlying the informality, yes.tim wood
    Not in direct opposition to this statement, but just a remark...

    Some educators are stuck on introducing maximum controversy when they present their students with novel abstractions. Expressions such as "as many as", I think, have to be reserved to already familiarized audiences, but some teachers find it particularly amusing to thrust the "inside jokes" of mathematics directly to the uninitiated.
  • The bijection problem the natural numbers and the even numbers

    From a programmer's point of view, cardinality equivalence between two sets is about their expressibility through each other.

    card(N) = card(E) tells me that if I want to define an encoding of the elements of E through the elements of N, or N through elements of E, I can. Thus, I can write/store n to represent "the n-th even number", and 2*n to represent "the ordinal n". In any case, as long as I standardize my representation upfront, I am going to be able to decode it uniquely later.

    card(N) < card(R) tells me that I can never represent a real number through a natural number, no matter how I try to tip-toe around the issue. I can muster the strangest encodings, but it is logical impossibility. Thus, I will be able to express only some real numbers ordinally (or even out of order), but not all of them.

    P.S. Thus your dilemma can be rephrased as the ability to encode some set and a strictly smaller set through the same representation set. This may be awkward in some sense, but it is just a fact of life.
  • Sorry for this newbish post.
    My interests are quite light and superficial compared to other forum posters. Still, I recommend the Bryan Magee interviews in philosophy as an accessible and general background.