KhakiHeron
I believe we can find a mechanism to represent complex values as single values - and I agree it likely is related to vectors. The example you give, however, remains with real values (x,y) and not a single value z - which could be thought of as the scalar value of the vector (which cannot end up as x+iy, as this is two real values and not a single complex value).

AquamarineJellyfish & CornflowerCicada
Whole numbers are abstractions - what is common to all single objects, what is common to all pairs of objects. Much of mathematics uses abstractions - what is common to all algebraic equations of the second order. Geometry can be more or less the same - what is common to all line segments, what is common to all squares, to all triangles. The geometry, arithmetic, and equations of math are statements about any object or concept that fits the abstraction. The equation of a pendulum is very similar to that of a vibrating string, or of a wave. The equation is about any abstraction of the concepts - which is why the pendulum equation can be applied to the pendulum, the vibrating string and a wave. Each physical system can be abstracted so that it fits the equation (maybe with some additional values or constants). The fact that the equations need to be tweaked to fit the reality is because no physical event exactly matches the abstraction.
The power of mathematics comes from this ability to abstract - and the equations can apply exactly to the abstracted situation. When applied to physical events, we can take an event and find that it is like the abstracted situation. As with the pendulum, string, and wave, we can find multiple events that abstract to the same (basic) equation. There are always differences between the physical event and the abstracted situation - always (we never get the exact values for every event that is represented by an equation). So mathematics is good at abstracted situations, it is only good to some degree of error when applied to a physical event.
Then there is the consideration of the symbols used for values, for the equations - for being able to communicate an abstract concept between people. This includes the numeric representational system (e.g., Roman numerals, ratios, positional real numbers, powers, logs), infinitesimal symbols (e.g., dx/dy, Newton's symbols, epsilon-delta), vectors, or matrices - and other mathematical symbols. These can vary significantly between societies, cultures, aliens - as they are devised creations not inherent to some deterministic appearance. Mathematics cannot progress without these symbols - and mathematics has been prevented from advancing without the introduction of new devised symbols. Our calculus would be more difficult using Newton's symbols than Leibniz'. We could not have the technology we have using just Roman numerals or just fractions.
Science and mathematics are very co-dependent, although there are few abstract situations (aside from simple arithmetic e.g., 1apple+1apple=2apples) that exactly always matches physical events. And the inexactness, or error terms, leave wiggle room for interpretation of which equations (or tweaks) fit the experimental results.
Science has a number of historic paths that have not taken us down the always 'correct' (in hindsight) path. So I do not see science as deterministic (nor do I see mathematics as such - for symbolic reasons).

    Lorraine Ford So- OK Here's the thing. IF we assume mathematics to be closed and separate from the real world, then we should have no concept of number whatever. And maybe that's true. Anyway, the properties of numbers are indelible in the roots of physics. You can't have an equation without them. Even if they're only approximate.
    Thank you for your helpful discussion CornflowerCicada Country folk know its a lot more economically efficient to undersell ideas and things, rather than oversell them. A lot of these 'colors' seem to represent something more variegated than some mere "hue." JBTW.

        Donald Palmer

        “Whole numbers are abstractions”

        But where do these abstractions exist? Whole numbers, rational and irrational numbers, mathematics, and all the associated symbols, only exist from the point of view of the human imagination. This is not to deny the “power of mathematics”, where mathematical symbols can be used to represent the real-world numbers and the real-world relationships that physics has obtained and deduced from experiments (despite some degree of error in the numbers).

        Re the counting numbers: the number of planets in the solar system is not an actual real-world property (“what is common to”) of a collection of planets, a property that actually exists “out there” in the world. The number of planets in the solar system is something that only exists from the point of view of the human imagination. When analysed, counting is a many-step process, performed by people and some other living things, which also involves: 1) the ability of the counter to define the category being counted (e.g. first, define a planet); and 2) the ability of the counter to identify the category being counted (e.g. second, does a particular celestial object fit the agreed-upon definition? Is Pluto a planet?).

        But, as opposed to what exists in the human imagination, there are numbers that actually exist in the real physical world “out there”. When particular categories of the real physical world are measured, e.g. relative position or mass, one gets a number, and this number is represented via the use of a number symbol. So real physical world numbers exist “out there” as opposed to the numbers that only exist in the human imagination.

        “The power of mathematics comes from this ability to abstract”,

        i.e. the power of mathematics comes from: 1) the human ability to invent and comprehend symbols; and 2) the human ability to represent the real physical world, and even imaginary concepts, with symbols. So, I guess that I agree with you about symbols.

          Charles St Pierre
          I never “assume[d] mathematics to be closed and separate from the real world”. I’m just trying to say that there are things that only exist in the human imagination (like binary digits, which can’t be measured), as opposed to what exists “out there” in the world (like voltage, which can be measured).

              CornflowerCicada
              I am not really a Platonist that believes abstractions exist outside of ourselves (the shadows on the cave wall that only shadows real existence outside the cave). I believe abstractions require some form of consciousness ('human imagination' works). This would apply to 'mass', 'distance', and other "physical world numbers" you refer to. These are all abstractions about the real world and so do not 'really' exist in the real world. We represent measurements about these concepts using symbols for the abstract concept of number.
              I very much doubt 'mass' was a concept two thousand years ago. 'Weight' was likely a concept back then, but not the abstraction to 'mass'. So I fail to see how such abstract scientific concepts can be considered existing 'out there in the real world'.
              It is quite an amazing thing that human imagination can generate a mental (some say 'internal') model of the world we perceive around us. Unless our minds work like the tents and suitcases in Harry Potter stories (that can be physically bigger inside than out), we build a mental model of the world we perceive that somehow fits into our minds. I find this only possible if the model is built with abstractions. And nothing of our abstract model exactly matches everything about the physical objects and reality we perceive.
              This is another reason why mathematics cannot truly represent reality - it can, at best, only represent our mental abstract model of reality.

                Donald Palmer
                Yes, the mass and distance categories, and the numbers we use to represent them, are just words and symbols that we use to represent/ model something that exists “out there” in the real world.

                But, in a world of “fake news” and “alternative facts”, fake realities that only exist in people’s minds, can science/physics be different? Can physics at least say what are the genuine characteristics of what exists “out there”?

                I think that one characteristic is that the world “out there” is indeed inherently categorised, because logically, a coherent physical world can seemingly only exist if interconnections exist: categories that are connected via relationship to other such categories. But just saying that a category that we would label as “mass” exists, that seems to be genuinely related to other such categories, doesn’t tell you anything much: one needs actual numbers to make things more specific, numbers that we would represent with number symbols. Another consideration is that people and other living things, with their conscious imaginations, are a part of, not separate from, what exists “out there”.

                You say: “I fail to see how such abstract scientific concepts can be considered existing 'out there in the real world'”. But would you say that there are any genuine characteristics of what exists “out there” in the real world?

                    Lorraine Ford
                    Do I think that human understanding incorporates everything about anything 'out there'? I would say No. We hold an abstract model of what is 'out there', which only includes some characteristics of what is 'out there'. I would say that all categories we deal with are of human construction that deal with human questions and perspectives. The categories are dependent upon the questions asked and the perspectives involved. If I am considering the taxonomic categories of living entities, I am likely considering species, genus, family, order, class, phylum, kingdom, domain. If I am considering categories of people with pets I like, then these taxonomic categories are not very useful.
                    If you have ever attempted to build a file-folder structure to organize files, the categories and structure are very dependent upon the questions being asked (do I structure conference presentations by time&date, by topic, by association hosting the conference?) If I am only concerned with the force an object subjects the ground to, I need only concern myself with weight. If I consider accelerating a spaceship to the moon, then I might be better off considering mass.
                    What of all the categories 'out there' that we have not yet identified? Do they exist 'out there' now?
                    To answer your last question - I consider 'characteristics' something different than 'categories'. Categories can change based upon the question and perspective of the person involved. Characteristics tend to remain the same even from different perspectives. However, even characteristics have their limits, especially if we move up or down in scale relative to an object. Measuring the area of a table is straight forward at our scale. It becomes a different matter at the molecular or atomic scale. The same can be said of many characteristics at our scale (even mass). So scale introduces different perspectives - even different objects - at different scales.
                    Can we measure the distance between the surface of the table and a molecule of a pen sitting on the table? Do these characteristics make sense when we cross many levels of scale? Many of the characteristics we have identified only work in certain situations and/or scales.
                    I will note that if we truly live in a three dimensional physical space, then this distance measurement should be easy and 'characteristics' should only involve levels of accuracy across scale (not what we have found, with tissue, cells, proteins, molecules, atoms, etc.). Consider that scale introduces a direction of space that different objects exist along (even at the same 3-D position), which changes the concept of a characteristic at one scale or another.

                        You seem to get physical determinism from mathematical determinism. But mathematical theories can be stochastic. Probability is a mathematical concept. Sure, 2+2 is always 4, but why can't mathematical probabilities make the physical world uncertain?

                        Donald Palmer
                        Thanks for your reply. You say: “all categories we deal with are of human construction that deal with human questions and perspectives”. But right now in 2023, it is apparent that a lot of nonsense, and fake ideas and fake questions about the world, occurs in the human imagination. So, on what basis can you say anything at all about the world? What are the characteristics of what actually does exist, as opposed to the potentially fake ideas that exist in the human imagination?

                        I agree with you that what doesn’t actually exist in the real world are the symbols we use to represent the world. Though symbols are physical (i.e. written on physical paper; spoken using sound waves) the “symbol” part of a symbol only exists from the point of view of the human imagination.

                        But I contend that what does actually exist is: 1) consciousness including the human imagination; and 2) as opposed to the human imagination, physics has shown that the real world out there is inherently categorised, and that these categories are interrelated, and that there are aspects of the world that can be represented by number symbols.

                        How can you talk about things labelled “distance” and “scale”, when you don’t first acknowledge that:

                        1. A distance (relative position) characteristic might exist in the real world as a genuine category of reality.;
                        2. A distance characteristic might be inherently related to other characteristics of the real world (related in a way that can be represented mathematically);
                        3. The number symbols acquired from distance measurement might represent a real-world number, as opposed to the numbers that might exist in the human imagination; and
                        4. The number symbols acquired from distance measurement can only represent a real-world number, IF the number applies to a genuine real-world category and a genuine real-world relationship?

                        Does a direct relationship pathway actually exist between the large and the small scale, so that distance between the large and the small scale would have a genuine meaning? Or is there only smaller scale mass/ relative position/ charge/ etc. infrastructure that, logically conjoined, leads to the larger scale?

                            Lorraine Ford
                            You appear to use the terms 'categories' and 'characteristics' in essentially the same way - so we appear to have different definitions (I do not use them the same way). So if I take your use of 'categories' to also mean 'characteristics' then when I use 'characteristics' you can also use 'categories' (although that does not work for me).
                            Assuming I am (mostly) correct on your use of terminology, then we appear to be saying similar things. We agree about 1) for sure. For 2) we can agree that there is a real world 'out there' that has characteristics (or your 'categories'). I would ask how 'physics has shown' anything about the real world out there? My response would say agreement with others is what has shown a real world exists 'out there'. Neither physics nor science can claim this of themselves, as I think the concept of an agreed world 'out there' significantly predates any concept of science. I would suggest that science grew out of an assumption that we can perform measurements and experiments on the world out there where different people would agree upon the results (the world out there being an assumption, not a demonstrable fact). This does not remove the abstract nature of our model of the 'world out there', nor of the mathematics we have devised to measure and experiment on this abstract model. And science does attempt to check our abstract models against our experience of the world out there. An adherence to checking model against 'out there' is part of the reason why science is a better methodology than mysticism or religion. The methodology does have limitations for singular events and internal subjective experiences.

                            The 'fake news' aspect comes from agreements between people that are not cross-checked against the world out there (or worse, the agreements are intentionally propagated knowing most people will not cross-check the statements). Good journalism and good science both have this cross-checking in common - 'fake news' does not or cherry picks the results to agree on. However the characteristics (or categories) are still human dependent - since they are defined by our sensory perceptions, which to date have a scale dependence on those perceptions.

                            I will agree that relative position implies, to a human, a distance between objects - however 'distance' implies a (human) measurement, while relative position does not. So they are not entirely equivalent concepts. Measurement requires a conscious mind and abstract model, while relative position can exist 'out there' as a characteristic.
                            So, while I can agree that the world out there has characteristics, the ones we measure are still human defined.
                            So I think I am saying that anything that we can measure and has mathematical representation (which are abstractions) presumes the abstract models we have in our minds and is not really in the world out there. With that said, in our daily lives we pretty much conflate the world out there with our abstract model of it and pay little attention to the abstractions and differences. And if most people (or most scientists) agree on an abstract model, it will be rather difficult to change that abstract model - even if with a better one.

                            Finally, if you are suggesting that there is "only smaller scale mass/ relative position/ charge/ etc. infrastructure that, logically conjoined, leads to the larger scale", then you have entirely bought into the abstract model and are no longer concerned with what your sensory perceptions tell you. There are articles that state 'You never actually touch anything, since atoms never actually touch' - cross-check this with your eyes touching another person or a window pane. If levels of scale fall along a distinct dimension, then both can be true - each at their own level (however this presumes a fourth spatial dimension - since different actions at different scales can occur at the same three-dimensional position). How do you explain how our scale humans created the LHC that manipulates the small scale? Does it all start with the cause at the small scale up to our scale and then back down to the small scale? How does this occur? Doesn't this also presume a dimension of scale that the cause moves up and down along?
                            I would go so far as to say that relative positions and actions at different scales, even appearing to be at the same position from the point of view of one scale, are characteristics of the world out there. Being able to measure the distance between these positions appears to be a limitation of our models and/or mathematics, since our mathematics is currently unable to provide a single value measurement of such a distance at this time.

                                Donald Palmer

                                “… if you are suggesting that there is "only smaller scale mass/ relative position/ charge/ etc. infrastructure that, logically conjoined, leads to the larger scale", then you have entirely bought into the abstract model and are no longer concerned with what your sensory perceptions tell you.”

                                The information we acquire via our senses is pretty basic low-level stuff, e.g. light wavelength. This has to be built into a bigger picture by the senses and/or the mind, and only then could it become “what your sensory perceptions tell you” about oneself and the surrounding world. This bigger sensory picture is seemingly not the numerical result of a big equation, but the logical conjoining of a lot of smaller fragments of information. This seems to indicate that there exists an aspect of the world that can only be symbolically represented by logical connectives. And I think that you already assume that a very fundamental thing called “logic” must exist.

                                “agreement with others is what has shown a real world exists 'out there'”

                                I think that, as opposed to the fantastical ideas that can exist in the imagination, in many ways physics has correctly modelled the real world “out there”, e.g. the missions that have been sent to Mars confirm that the modelling is pretty good. So, using logic, I contend that, as a result of physics’ research and modelling, the 3 essential types of information that are seen to characterise the real physical world are categories like mass and momentum, relationships between these categories, and the numbers that are associated with the categories. (By contrast, a world couldn’t be built out of numbers alone because a lone number without an accompanying category, and without an inherent relationship to other such categories, is entirely useless and meaningless as a source of information.)

                                “Measurement requires a conscious mind and abstract model, while relative position can exist 'out there' as a characteristic”

                                I disagree. Because it is not enough for the 3 basic, essential types of information needed to construct a physical world (categories, relationships, numbers) to merely exist: for the world to function, there must also exist a knowledge component, whereby the world knows these essential aspects of the world (panpsychism).

                                Re small scale and large scale and scale in general: I think it all depends on first investigating how the world is constructed and held together, especially when it comes to the case of living things.

                                  Lorraine Ford The human imagination exists in the universe. Human imagination is limited to the forms and structures available to that universe. we cannot imagine what we cannot imagine, and we cannot imagine what the universe, what our experience does not provide reference for. I challenge you to examine your imaginary worlds closely.

                                  The numbers, numbers and their prime factorizations, are those parts. And the universe is all the arrangements and relationships between those parts.
                                  Everything is either number, (Mathematical monism,) or everything is number and something else not number. Or everything is not number, but mathematics seems to be "unreasonably effective." How can I describe it if it's indescribable. How can we account for its action, if we cannot describe its action? And if we describe its action with number ,how is it necessary to postulate this other non-number part.?
                                  I mean, it didn't work for God. Why should it work for anything else?

                                      Charles St Pierre
                                      There is an awful lot of nonsense that can exist in the human imagination. For whatever reason, in the human imagination, 2+2 = 5 is a concept that can exist. What exists in the human imagination is not the same as the real measurable physical world “out there” outside the human imagination, where 2+2 cannot = 5.

                                      In the real measurable physical world, numbers, that are not associated with a category/ relationship, are totally meaningless and useless, and can’t even exist. The prime factorisation of numbers is something that human beings do: the prime factorisations of numbers are not things that exist in the real measurable physical world: they only exist in the human imagination.

                                          5ive+5ive e=qual 2wo --------------- hands 👾

                                          other animals can do the same elementary arithmetics maybe arithmetics is universal, maybe there is a common ancestor

                                          for the far milenium years+ future -knowing how to count will be not that different with other actions like eating hot soup , or doing a sculpture in marble .

                                          in a more practical down to earth example
                                          in french numbers up to 100 have a particularity of being less obvious to name , this might explain the high number of french mathematicians, maybe they were annoyed by that and spent extra effort to understand , with the benefits / reward, this effort made them be mathematicians.
                                          in arts a portrait picture or paint is more realistic if its containing an ugly memorable thing (particularity/ speciation /sofistication)

                                          (maybe) what is needed is( concerning early education ) to teach the next generation of people to do mathematics(not only stem ,and humanities) in some weird particular , not for efficient necessity, unique way , with different names,and absourd rules , that the previous generation would look at it to be unexpected, that would increase creativity and nurture the evolving language
                                          going to an extreme such that when two mathematicians in the year 2123 of the same field educated in different "countires" meet they must spend extra time to understand each others . (to refactor the myth of math as common language, maybe it should be a reason for language)

                                          i haven't started to look in to much in details at the Kurt Goedels work, what i have remained with, so far, is that, at the outside mathematics looks like a solid strict construction , although looking closely (well not me, this is how i make the comparison) where there is a fine sieve that can be made in to portals and the construction evaporates in thin air . or get trapped/tracked in a flexible maze

                                          You have to think big!
                                          It is known that Newton determined the gravitational coefficient through the parameters of the orbits of the planets of the solar system. If the gravitational coefficient is determined in a similar way from the parameters of the orbits of electrons in the Hydrogen atom, then the gravitational coefficient of the planetary system of the Hydrogen atom becomes 40 orders of magnitude greater than in the solar system. Then the Planck parameters of the Hydrogen atom are the parameters of an electron with its radius equal to the radius of the Compton wave of the electron. Those. each level of fractal matter has its own “Planck parameters”, and the generally accepted Planck parameters are an abstract delusion and have no real meaning at all. Indeed, what relation does the gravitational coefficient from the parameters of the Solar system have to the parameters of the planetary system of the Hydrogen atom? None!!!

                                          You have to think big!
                                          The fine structure constant can be easily calculated with an accuracy of up to 7 digits, assuming that all elements of matter have a fractal structure. Then, therefore, "black holes" do not exist, and there is no event horizon. Those. inside putative "black holes", there is deterministic matter that obeys the simple quantum laws of fractal matter, which unify gravity and quantum phenomena of the deterministic functioning of matter on all scales of the universe [ appendix: https://s3.amazonaws.com/fqxi.data/data/essay-contest-files/16/reference_id_2304.pdf
                                          https://qspace.fqxi.org/competitions/entry/2304#control_panel ].

                                          Lorraine Ford Lorraine Ford Well, see, I understand how 2+2 =4 (Also 2 x 2 =4. 4 is the only number to do this. An amazing degeneracy. Might have something to do with Fermat's Last Theorem. Some speculation, that. Anyway. So this 2 + 2 = 5 thing. Describe how it happens. What are the details, the picture, the process, how this happens, in your imagination. There must be some sort of math behind it. And if there is, then it describes something which the math which describes this universe describes. Assuming 'math' to be an object which is connected. If, of course, math is composed of components which are not connected, then how could we discover one component from another? Except by some connection. Unless your imagination is not connected to the mathematics of the universe. That could be. But if it is connected, then it can only be a part of the mathematics.. And thus limited by its limitations. By the limitations of numbers and number theory. But that's the way I see it. We just haven't figured out how the pieces fit together, so the possibilities seem endless.
                                          But It is a non-sequitur to imagine that because we cannot realize a creation of our imagination beyond the possibility of realization in this universe, that there exists some universe where we can.,

                                                Charles St Pierre
                                                Do you deny that a lot of nonsense can exist in the human imagination, nonsense that doesn’t exist “out there” in the real physical world? E.g. the nonsense idea that the earth is flat might exist in the human imagination, whereas in the real physical world “out there”, the earth is almost spherical.

                                                Re “numbers and number theory”: you can’t build a universe out of numbers alone, because numbers have no inherent categories or relationships. In the real physical world, information comes in the form of: real-world categories like relative position, wavelength and mass; real-world relationships between these categories; and real-world numbers (that apply to these categories) that are obtained by measurement.

                                                I never suggested that “because we cannot realize a creation of our imagination beyond the possibility of realization in this universe, that there exists some universe where we can”. I’m not sure where that sentence of yours came from!

                                                  Write a Reply...