Can someone explain this to me?


Peter Boghossian blurted this out tonight.


There are no right angles in nature, yet no one says right angles are *social* constructs because they’re not morally motivated to do so.

There are right angles in nature. We also have social constructs built around ideas about right angles — look, Boghossian just made one, stating an idea about right angles and the nature of our interactions with them. I am baffled and have many questions.

  • Is he drunk-tweeting?

  • Does he have some point that he is trying to make, poorly and insipidly?

  • Is he so annoyed that humans are social animals who build mental models of how the world works,
    and that the map is not the terrain, that he is lashing out in defense of some kind of Platonic absolute?

  • Is he a very bad philosopher?

  • Is he not very bright?

A lot of the people I follow are currently rather flabbergasted at this flaming nonsense.

Comments

  1. Hj Hornbeck says

    Wow, Boghossian doesn’t know his physics. Photons are electromagnetic waves, where a change in the electric field induces a change in the magnetic field and vice-versa. The motion of those fields are precisely ninety degrees to one another, and this flows directly from the physics of nature.

  2. rayceeya says

    “Right angles in nature,” makes me think of Giants Causeway, or halite crystals. I’m sure there are plenty of other examples.

  3. consciousness razor says

    He might be thinking of a perfectly formed cube or some geometrical figure like that, not angles or distances themselves. (The latter would be wacky; space doesn’t lack those.) And it wouldn’t do to have items located at the vertices, since you need the edges and faces, maybe it’s also a solid … not to mention a system of units or other conceptual tools to make sense of these things. There’s some very specific thing like that, which doesn’t exist as an object in the natural world, but is only something people analyze in mathematical terms.

    That’s probably what his dumbass point is supposed to be about. Theories exist, and no one says “people make theories.” That seems to be the gist of it.

  4. slithey tove (twas brillig (stevem)) says

    It’s some deep metaphor that is beyond comprehension. “Right angle” might be using “right” in the “morally correct” sense. “There is no correct angle” as it is all a moral construct and no absolute definition of morally correct. “Angle” being a synonym of “approach”.
    Argh getting over my head…

  5. doubtthat says

    Maybe I don’t understand the twisted path of reasoning that leads to their meaning of “social construct,” but wouldn’t the fact that right angles don’t appear in nature be evidence that they are a social construct? I get that he was (unsurprisingly) wrong on the facts, but those examples seem to strengthen the dumb point he was trying to make.

    Unless there are lava-vent bacteria I’m unaware of engaging in theoretical mathematics, I’m pretty sure the further math moves from the empirical, the more socially constructed it becomes.

  6. says

    I think Boghossian is correct. Most/all things are socially constructed, at least in part, but we don’t talk about the social construction of X when we’re not motivated to do so. Generally, it is correct to not do things that we are not motivated to do?

    Except that right angles exist in nature.

  7. John Harshman says

    The phrase “social construct” leads me to suspect he’s trying to make some kind of point about race. But I don’t know what kind of point.

  8. John Morales says

    Maybe he’s read a book recently; Peter Watts’ vampires have the “Crucifix Glitch”:

    […] a cross-wiring of normally-distinct receptor arrays in the visual cortex, resulting in grand mal-like feedback seizures whenever the arrays processing vertical and horizontal stimuli fired simultaneously across a sufficiently large arc of the visual field. Since intersecting right angles are virtually nonexistent in nature, natural selection did not weed out the Glitch until H. sapiens sapiens developed Euclidean architecture; by then, the trait had become fixed across H. sapiens vampiris via genetic drift, and – suddenly denied access to its prey – the entire subspecies went extinct shortly after the dawn of recorded history.

  9. Mark Jacobson says

    Sounds like he’s mad about some subset of people being treated with respect, and he’s making this absurd analogy to try and discredit social construct based arguments.

  10. jahigginbotham says

    @4 rayceeya
    Halite yes (cubic); causeway no (only crudely hexagonal due to uniform slow cooling and low energy form).

  11. says

    @#11, doubtthat

    Unless there are lava-vent bacteria I’m unaware of engaging in theoretical mathematics, I’m pretty sure the further math moves from the empirical, the more socially constructed it becomes.

    I deny that any portion of mathematics is actually empirical. The real world doesn’t even have a decent analog to integers in the mathematical sense unless you go into physics which wasn’t discovered until the 20th century, long after mathematicians had moved far beyond that. (Counting objects? Show me two objects which are completely identical, or three objects where the differences between a pair of them are identical to the differences between a different pair.) As far as we can tell, reality follows mathematical rules because — at the lowest levels — reality is composed out of entities which might as well be mathematical constructs, not because we constructed mathematics to match reality.

  12. Ed Seedhouse says

    Aside from airy invisible “things” like lines of force and vectors and such, the most obvious example I can think of is salt crystals, which are cubical. Cubes involve right angles last I heard. Of course the ancients did not have magnifying glasses and their eyes might not have been sharp enough to see them. I can’t think of any big obvious things that are naturally cubical or involve right angles, but I may be (probably am) missing something obvious. Polarized sunglasses were not available to the ancients either or so I have heard.

  13. consciousness razor says

    Vicar, I agree that math isn’t empirical, but this argument is wrong:

    Counting objects? Show me two objects which are completely identical, or three objects where the differences between a pair of them are identical to the differences between a different pair.

    You can count objects which aren’t identical. No matter how different they may be, they are nonetheless countable objects. The identity of indiscernibles is alright as a principle, as far as it goes. However, it’s not written in stone that you must only count identical things. Indeed, if two apparent things were actually the same thing, then the conclusion is that there is only one. But you can still count that — it isn’t the same as zero — and the counting numbers are off to the races at the point.

    Anyway, I’m sure you’re not going to deny that there are at least two objects. Even if you wanted to press the argument…. Well, there’s me, and then there’s you. We’re evidently disagreeing, and we add up to two nonidentical things. It’s not as if you could just wish that argument (or me) away.

    As far as we can tell, reality follows mathematical rules because — at the lowest levels — reality is composed out of entities which might as well be mathematical constructs, not because we constructed mathematics to match reality.

    Now I’m lost again. If it’s not empirical, then I don’t get what part of logical space you’re trying to occupy here. Physical objects are not composed out of mathematical abstractions. The latter are not located anywhere or at any time, do not consist of any material, etc. There is no way to cram such features into a fundamental reality which simply don’t have them, in order to get a recognizable world out of it. (I mean, sure you could make a totally disrespectable claim and just say so, but it would be manifestly arbitrary and the conclusion is absurd. That’s what I mean by saying there’s way to do it.)

    Math works for us (to represent reality, not be reality), because it’s logical. This isn’t to put logic at the top of the pile, but to say it’s more general: you are thinking in a logical way when you think mathematically. It’s a specific way of doing it that makes direct use numbers, geometry, patterns, and so on. The world is certainly confusing for us at times, but it doesn’t do things like contradict itself. It’s trivially true that we live in a logically possible world, not an impossible one. Since that is true, some kind of math will in one way or another be capable of representing such a world. In that light, its success is not a miracle at all or something that requires some deep explanation that you can’t even articulate — it’s exactly what you ought to expect.

  14. says

    @18: “Of course the ancients did not have magnifying glasses and their eyes might not have been sharp enough to see them.”

    It’s fairly easy to grow salt crystals that are not only discernible to the naked eye, but are quite large. And large halite crystals exist in nature, as do many other types of crystals. Ancients (educated ones at least) surely would have been aware of them.

    I’m also having a hard time thinking of larger scale natural things that are at right angles, mostly because when you get above the scale of molecules arranging themselves in a specific pattern, things are often too messy to get a precise right angle. However, many plants have secondary stems that form roughly right angles with the main stem. Trees usually grow at roughly a right angle to level ground. Human incisors are roughly rectangular when viewed from the front. Water falling off a cliff does so at a roughly right angle to its source. Come to think of it, most of these examples are due to gravity pushing things down perpendicular to level ground, something that people would have had an intuitive understanding of even very long ago.

  15. says

    @consciousness razor

    You can count objects which aren’t identical. No matter how different they may be, they are nonetheless countable objects. The identity of indiscernibles is alright as a principle, as far as it goes. However, it’s not written in stone that you must only count identical things. Indeed, if two apparent things were actually the same thing, then the conclusion is that there is only one. But you can still count that — it isn’t the same as zero — and the counting numbers are off to the races at the point.

    Real-world macro objects do not have the qualities which are necessary for mathematical objects to have, and the mathematical objects certainly predate any attempts to define them in terms of real-world objects. Integers have to be precisely spaced, and real-world objects aren’t. If you have a bag of rocks, you can count them — but I can smack the bag with a hammer and change the result. If you start quibbling about other things — let’s say you’re counting chickens, for a very popular example — is a chicken with a serious injury (like a leg missing) a whole chicken? Is a chick a chicken? Is a fertilized, soon-to-be-hatched egg a chicken? When, precisely, does it become a chicken? To use real-world objects to define “counting” means having to answer a whole lot of weird questions which simply don’t apply to integers (and if those questions did apply — if we had to ask “is this actually a number, or not” for each one — then mathematics would be impossible).

    If you like Platonism (and can ignore the infinite regressions involved), then: the act of counting is the act of drawing a comparison between the integers and groups which have some shared quality to them, but you will never actually encounter an integer.

    Physical objects are not composed out of mathematical abstractions.

    You can’t prove that, actually. The subatomic realm behaves very much as though it is a specialized mathematical construct. You can’t demonstrate that this is not true (at least, on currently-available evidence) — but any claim that “reality” is something else had better include an explanation for why reality behaves like a mathematical construct when you look at it very, very closely, or else that claim is simply wrong. (It’s like the occasional nutter who wants to claim that, to use an example I saw a decade or so back, there’s another electron shell with energy below 1s, and we can get energy out of atoms when the electrons drop to this lower-energy state. In order for that to be true — news flash: it isn’t — then there would have to be an explanation for why electrons aren’t already observed to be dropping into that state, and emitting characteristic photons as they do so.)

    If it’s not empirical, then I don’t get what part of logical space you’re trying to occupy here.

    If mathematics were empirical, it would mean that mathematics was derived from physical reality. It isn’t; in some cases demonstrably so. I am asserting that, if you are going to claim either that mathematics was derived from a non-mathematical reality or that reality derives from mathematics, it is at least plausible that the latter is true: when you get down to the smallest level, everything is ultimately fluctuations in “fields”, which means simply that numerical values are changing in patterns. Sure sounds like a mathematical construct to me! Ultimately, we are built out of numbers, like it or not.

    The latter are not located anywhere or at any time, do not consist of any material, etc.

    Tell me the precise location of a photon, or what material an electron is made of. Once you get down to that level — which is the only level that counts for this sort of discussion — location is merely a property (and one which is somewhat poorly defined, at that) and material dislimns into waves, which consist of numerical values fluctuating.

    There is no way to cram such features into a fundamental reality which simply don’t have them, in order to get a recognizable world out of it. (I mean, sure you could make a totally disrespectable claim and just say so, but it would be manifestly arbitrary and the conclusion is absurd. That’s what I mean by saying there’s way to do it.)

    I don’t have to cram such features into a fundamental reality, because quantum physics has already done it for me.

    The world is certainly confusing for us at times, but it doesn’t do things like contradict itself.

    Strictly speaking, we do not know this. We know that there have been no physical contradictions which we have been capable of detecting, within the span of our recorded history, which is… well… a dot on a certain thread. But, not to put too fine a point on it, we not only can’t be sure it will never happen, we don’t even really know what such a thing would look like. Although I am inclined to agree that the universe does not contradict itself, contradictions may simply be black swans — heck, if you had asked a physicist in 1989 what a physical contradiction would look like, they might have answered “if the universe’s expansion were accelerating, it would be a contradiction”, and yet here we are with dark energy waiting to be explained.

  16. jahigginbotham says

    @22 vicar
    The subatomic realm behaves as a mathematical construct but macroscopic objects do not have the qualities which are necessary for mathematical objects to have? Is it silly to ask why this doesn’t seem to make sense? Where did this extra complexity (or whatever it’s called) come from?

  17. says

    @The Vicar,
    Not sure what you’re going on about, but elementary particles are literally identical. I take it you haven’t reached that part of quantum mechanics yet.

  18. consciousness razor says

    Real-world macro objects do not have the qualities which are necessary for mathematical objects to have, and the mathematical objects certainly predate any attempts to define them in terms of real-world objects.

    What do you mean by “predate”? Abstractions don’t exist in spacetime.

    Integers have to be precisely spaced, and real-world objects aren’t.

    Precisely spaced points are precisely spaced. There doesn’t need to be a macro-object that you’re representing with distances or whatever measure. Maybe there’s a pair of field values in those locations, if you really think you need them,* but even without it, there’s certainly nothing impossible about “spacing” in reality.

    *But that goes against the whole point of saying math (the practice of it) is not empirical.

    If you have a bag of rocks, you can count them — but I can smack the bag with a hammer and change the result. If you start quibbling about other things — let’s say you’re counting chickens, for a very popular example — is a chicken with a serious injury (like a leg missing) a whole chicken? Is a chick a chicken? Is a fertilized, soon-to-be-hatched egg a chicken? When, precisely, does it become a chicken?

    So we can come up with stupid or inaccurate or confused ways of representing the world. But there are better ways too.

    To use real-world objects to define “counting”

    I’m not using them to define counting. (I thought we agreed that math isn’t empirical, so that should’ve been settled.) You still can count those things. That’s one way that you can use counting, not the definition of it and not using the procedure to somehow define itself.

    You can’t prove that, actually.

    It follows from the fact that mathematical abstractions aren’t located in spacetime, aren’t material, etc., while stuff in the world does satisfy such criteria. Something about reality has those features. If we only had abstract objects which lacked them, then it wouldn’t be a world with them.

    The subatomic realm behaves very much as though it is a specialized mathematical construct. You can’t demonstrate that this is not true (at least, on currently-available evidence) — but any claim that “reality” is something else had better include an explanation for why reality behaves like a mathematical construct when you look at it very, very closely, or else that claim is simply wrong.

    I have no clue what you’re talking about. Look around: as you yourself said, “you will never actually encounter an integer.”

    If fundamentally things are composed of objects (mathematical abstractions) that you never actually encounter, then fundamentally there must be things which are composed of stuff that you do encounter, because actually and undeniably you are encountering stuff. Don’t buy into the quantum woo bullshit people have been selling for ages — no discovery of physics could have changed that.

  19. jahigginbotham says

    @11 doubtthat
    Your point sounds good to me. Maybe he means they are a mathematical construct and not a social one? Maybe a good philosopher is like an oracle? vague enough to always be interpretable to coincide with reality?

  20. jahigginbotham says

    @24 Siggy,
    no, he left himself an out in 17
    “The real world doesn’t even have a decent analog to integers in the mathematical sense unless you go into physics which wasn’t discovered until the 20th century”
    But how do we know elementary particles are literally identical? Maybe the differences are just too small for us to detect?
    At one time, the smallest particle of a material was an atom and all atoms of an element were identical. Then maybe 100 years later, hey, isotopes.

  21. jahigginbotham says

    @22 Vicar
    “you will never actually encounter an integer”
    Is not the number of Nobel prizes i have won or unicorns i have met an integer? And if everything material is unique, haven’t i encountered 1. And then why can’t i count at least from 0 to 1?

  22. =8)-DX says

    @Hj Hornbeck #1 (and others giving examples of right angles in nature)

    The motion of those fields are precisely ninety degrees to one another, and this flows directly from the physics of nature.

    Isn’t that kind of precision broken due to curvature of space and other interactions those fields will have with surrounding fields and particles? Like, isn’t there a nonzero curvature of space everywhere alongside a nonzero electromagnetic field, as well as nonzero effects of virtual particles everywhere? The same would effect the “precision” of right angles in crystals (plus there you have the “positions” of electrons within the crystal lattice as well as presumably some internal structure of the nuclei)? I guess objects and fields moving in relation to one another through a set of angles around 90 might be said to have passed through 90 at some point, but doesn’t the Planck time kind of do away with such an analog transition including “precisely” 90?

    Ah, it’s great to read another pharyngula thread where people are lustily whacking each other over the head with physics and philosophy.

    =8)-DX

  23. consciousness razor says

    Is not the number of Nobel prizes i have won or unicorns i have met an integer?

    What you didn’t win and haven’t met are Nobel prizes and unicorns, respectively. Nobel prizes and unicorns aren’t numbers. Zero obviously is, but those things are not.

    Isn’t that kind of precision broken due to curvature of space and other interactions those fields will have with surrounding fields and particles? Like, isn’t there a nonzero curvature of space everywhere alongside a nonzero electromagnetic field, as well as nonzero effects of virtual particles everywhere?

    Curvy or not, angles are still well-defined in those types of spaces … at least in general, but I don’t know enough about it to rule out some exceptions that presumably aren’t terribly useful in physics.

    Whether or not spacetime points form a continuum is another matter, but that’s not a reason to think a specific quantity like 90 degrees is somehow off-limits. Maybe it just so happens that a big lump from 88 to 92 is off-limits, in every bit of the universe, since no matter what it’s not physically allowed for any three things to be configured like that. It would be sort of odd if anything like that were true (obviously it isn’t, since that range would be vastly too large to agree with everyday life). But we don’t have a quantum theory of gravity, which is presumably the only way you could have a result like that.

  24. rietpluim says

    If right angles are a social construct, then we can choose to construct them differently. Like, right angles are now any angle between 13 and 43 degrees… Hey, suddenly there are plenty of right angles in nature!

    </about-as-stupid-as-Boghossian>

  25. jahigginbotham says

    29 =8)-DX
    How do you define the right angles in a halite crystal? No real crystal will have exactly 90 degree angles due to impurities, growth issues, and other defects. Small parts of it may, but then there is thermal motion of the atoms as well as lattice vibrations. And as you say, do the electrons have to be in the same configuration for the symmetry related atoms?

  26. jahigginbotham says

    29 =8)-DX
    Perhaps not 90 degrees, but maybe still right angles? Does the curvature of space(time whatever) require the use of non-Euclidean-planar geometries?

    “One of the many ways of comparing these geometries and the planar Euclidean geometry is to look at the sum of the interior angles of a triangle in each of them. In the spherical geometry the interior angles always add up to more than two right angles (180 degrees), in the planar geometry they add up to exactly two right angles, while in the hyperbolic geometry they add up to less than two right angles.
    Here is an example of a triangle on a sphere, with three right angles (adding up, therefore, to 270 degrees).”

    http://www.math.cornell.edu/~mec/mircea.html

  27. chrislawson says

    Not only do right angles occur in nature, but from a mathematical perspective, space contains an infinite number of them.

  28. jazzlet says

    There are some kinds of rock that natually fault into cuboids, though how near they are to perfect I haven’t measured. I wondered if they inspired the building of walls with cut cuboids rather than with whatever the local rock produces naturally.

  29. inflection says

    @23 jahigginbotham There are a number of different aspects to the curiosity of how macro properties arise from atomic microstates. Here’s one part of the answer.

    Suppose I take a particle whose possible energies are a bell curve, most likely around some mean energy A but with a distribution of probabilities to either side. This bell curve has a certain width compared to its amplitude. It might look fairly wide.

    Now suppose I take two exactly similar such particles, and the measurement I am interested in is their average energy. Their total energy, divided by 2, will be just as large, but the curve will not appear to be as wide compared to the range because it is less likely that they are both any particular distance from the mean. So the curve looks thinner.

    With a million such particles, the bell curve has become so thin that it starts to look like a narrow spike. With a billion of them, the bell curve is practically indistinguishable from a zero-width blip at the mean. From a random ensemble of particles I have arrived at a sharply defined temperature for the macro object.

    …for the larger conversation, all I can say is that mathematicians are fully aware of what Boghossian describes, to the point that we joke that we are constructivists over coffee and Platonists at work. We don’t care because it doesn’t affect our lives. The fact that race or gender is socially constructed has an immediate and measurable impact on people’s lives when claims of biological race or gender traits get used to deny them equal participation in society, and so of course we’re going to talk about that more.

  30. vole says

    Integers are abstractions. “Two”, for example, is the property that all sets of two objects have in common. You can have two camels, with two humps each, but take away the camels and all the twoness has gone too.

  31. Dunc says

    no one says right angles are *social* constructs because they’re not morally motivated to do so.

    Well, yeah – that’s because nobody uses the concept of right angles as an excuse to discriminate against people, or to act in other immoral ways. So what?

  32. richardh says

    jahigginbottom@27

    But how do we know elementary particles are literally identical? Maybe the differences are just too small for us to detect?

    No. Indistinguishability implies Fermi-Dirac or Bose-Einstein statistics, rather than Maxwell-Boltzmann.
    Without F-D statistics you wouldn’t have chemistry; without B-E you wouldn’t have lasers.

  33. doubtthat says

    @17 The Vicar

    I agree with you (acknowledging that I’m not well-versed in the nuance of this particular philosophical issue).

    I was trying to put myself in the place of one of these goofballs enraged at the notion of a “social construct.” Maybe my total lack of a similar motivation (which is probably racism or sexism, as others have pointed out) renders me incapable of convincingly adopting their point of view, even for a thought experiment, but I just can’t make sense of his position either from my perspective or his.

  34. doubtthat says

    Should say, I agree that math is not empirical, with respect to how we’ve discussed it here.

  35. John Harshman says

    #41 agenoria

    White dead-nettles have square stems.

    So do all other members of the mint family.

  36. leerudolph says

    Hj Hornbeck @6 wrote:

    Oh yeah, and I forgot about eigenvectors. They represent vectors which remain fixed in direction after matrix multiplication. If a real-valued matrix has more than one eigenvector, they must all be 90 degrees from each other.

    The link you gave includes, in addition to the “real” part you quoted, the further hypothesis—very strong, and necessary for the conclusion “they must all be 90 degrees from each other”—that the matrix be “symmetric”. Lots of non-symmetric matrices arise in physics and engineering!

  37. says

    jahigginbottom@27
    You may be interested to learn about the indistinguishability of particles. It’s actually really important that electrons are exactly identical in all properties. That’s what causes the Pauli exclusion principle, which prevents large atoms from collapsing.

    =8)-DX @29

    Isn’t that kind of precision broken due to curvature of space and other interactions those fields will have with surrounding fields and particles? Like, isn’t there a nonzero curvature of space everywhere alongside a nonzero electromagnetic field, as well as nonzero effects of virtual particles everywhere?

    In a curved spacetime, everything is still flat on a local level. So you can talk about local vectors (like electromagnetic fields), but it’s difficult to define non-local vectors (like the displacement between two atoms in a crystal lattice).

    The really easy example of right angles in physics is the Lorentz force, and all those cross products in electromagnetism. But also, if you have two time-varying vectors, surely they will make exact right angles at some instant in time.

    Anyway, it’s not clear why the right angles have to be “exact”. “Right angles for all practical purposes” seems good enough–depending on the social context of the discussion!

  38. Ed Seedhouse says

    A “point” in our physical universe cannot be said to have a position except as an abstract idea. Elementary particles in quantum field theory are considered to be points because the math is easier to do (still very difficult though!) and the predictions made using this assumption are extremely accurate, so accurate that no observation has yet been made that contradicts the theory. But also in quantum theory these “particles” have wave properties, in fact all particles have wave properties, not just one dimensional ones.

    Are elementary particles really “points”? It doesn’t matter so far as the usefulness of the theory is concerned so physicists are not bothered about it. The math works with points, so leave it be until some observation contradicts the math. Then we’ll need to figure out a new theory.

    Countable objects don’t need to be identical, as pointed out above. Take rocks. I may have five rocks of different sizes, shapes, and colours, but I can count them and so could our ancestors because rocks are easily seen and distinguished from other “things”. But hidden in this ability to count rocks is the idea of a “set”. What I am counting is actually instances of the set “rocks”. This only becomes clear once we think clearly about what it means to count, and to do that we need to build an abstract theory. So it took some hundreds (thousands?) of years to come up with the idea of a “set”.

    But “rocks” are easy to see and easy to count.

    I am not sure if we need to point to natural instances of right angles to come up with the idea of a “right” angle. If we are interested in building a wall with the least amount of rocks, we will want it to take the shortest path possible, and voila! right angles.

  39. Hj Hornbeck says

    Adding on to Siggy, curved space-time appears flat to things embedded within it, much the same way as a balloon appears to be a 2D surface to a teeny ant crawling across it. If you could step outside of space-time, sure, you’d spot the curvature.. but then you’re not in the universe anymore, a physical impossibility. We can only infer space is curved by watching for curvature changes, such as what happens when a star moves across the sky.

    leerudolph @45:

    The link you gave includes, in addition to the “real” part you quoted, the further hypothesis—very strong, and necessary for the conclusion “they must all be 90 degrees from each other”—that the matrix be “symmetric”. Lots of non-symmetric matrices arise in physics and engineering!

    Ack, I missed that! Still, the important question is where those matrix coefficients come from; if they’re a fundamental part of the universe, then it doesn’t matter that non-symmetric matricies exist or even if they are common.

  40. Rob Grigjanis says

    leerudolph @45: The real symmetric requirement is just a special case of hermiticity. Much more interesting (e.g. Pauli matrices)! The real part of a Hermitian matrix H is symmetric, the imaginary part antisymmetric. All eigenvalues of H are real, and eigenvectors with distinct eigenvalues are orthogonal. In quantum mechanics, physical observables correspond to Hermitian operators.

  41. Rob Grigjanis says

    Ed Seedhouse @47:

    Elementary particles in quantum field theory are considered to be points because the math is easier to do…

    In scattering calculations, they’re idealized as plane waves (i.e. with definite momentum), so they are not considered as points in space at all.

  42. militantagnostic says

    Perhaps Boghossian has conceptual penis envy. Say what you will about about Freud, he makes good circular saw blades.

  43. Rob Grigjanis says

    Siggy @46:

    “Right angles for all practical purposes” seems good enough–depending on the social context of the discussion!

    Ah yes. Like my initial orientation at a social (vertical), compared to my final orientation.

  44. aziraphale says

    My guess is that he’s annoyed by trans people wishing to define “gender” in ways he doesn’t approve of.

  45. Ed Seedhouse says

    Rob Grigjanis@50: “In scattering calculations, they’re idealized as plane waves (i.e. with definite momentum), so they are not considered as points in space at all.”

    Well, I am certainly not up on the latest of quantum field theory, I hardly understand it at all except in a rather vague way, as I am math impaired (or maybe just math lazy). But I was talking, perhaps not clearly enough, of a particle whose wave function has collapsed so that it may be said to have a position, when as I understand things it is describes as a dimensionless point.

    (But not a one dimensional point as I said in my early message thus committing a contradiction. Shudder.)

  46. robnyny says

    When I was in college, I went to a concert where the composer had composed a piece of music that demonstrated that there are no integers in nature. Apart from things like H2O and NaCl, I asked him if his dog had ever given birth to a non-integral number of puppies.

  47. Rob Grigjanis says

    Ed @56:

    But I was talking, perhaps not clearly enough, of a particle whose wave function has collapsed so that it may be said to have a position

    OK, sure. The detection of particles still involves (more or less) definite detection points. My point was about the calculations (the math) in the theory. We’ll call it a draw! ;-)

  48. robro says

    Boghossian must be a genius. One of his specialties (per Wikipedia) is critical thinking. Look how much critical thinking he’s inspired here in a matter of a couple of hours. It looks like an interesting discussion.

    While I was missing the start of this, I happened to be researching what to call the fact that many customers of user documentation seem not to notice numbers at the beginning of each step in a process. I miss seeing them frequently. While the notion of using numbers to indicate an ordered sequence of steps is straight forward, it seems the behavior of people looking at such things is to miss them. I don’t know if that’s phobic, although sometimes it might be, but a more general numerical cognitive phenomena.

  49. chrislawson says

    Ed Seedhouse@56 — remember that according to the Heisenberg Uncertainty Principle, we can’t know a particle’s position with perfect precision (as this would imply literally infinite uncertainty in its momentum).

  50. chrislawson says

    H J Hornbeck@48:

    …curved space-time appears flat to things embedded within it, much the same way as a balloon appears to be a 2D surface to a teeny ant crawling across it. If you could step outside of space-time, sure, you’d spot the curvature.. but then you’re not in the universe anymore, a physical impossibility. We can only infer space is curved by watching for curvature changes, such as what happens when a star moves across the sky.

    Not so. If we were unable to observe spacetime curvature then the last century of physics experiments on general relativity would have found nothing. Furthermore, we observe the curvature of spacetime every waking second of our lives. It’s called gravity.

  51. Ogvorbis: Swimming without a parachute. says

    robro @59:

    I miss seeing them frequently. While the notion of using numbers to indicate an ordered sequence of steps is straight forward, it seems the behavior of people looking at such things is to miss them.

    I ignore those all the time when building fine scale models. Partly to make painting easier. Partly because (especially with the smaller companies) the directions, if followed step by step, lead to me having to remove a piece already glued in place so that I can put another assembly underneath it. I will not name the manufacturer of the Krupp 20.5 cm howitzer which had me tearing my hair out. So, insomnia cases, it may not be that they, er, rather, we, don’t see the step numbers, sometimes we ignore them intentionally.

    And, Boghossian, as far as right angles go? I’ve seen many rocks and minerals with beautiful right cleavages. And I’m a liberal arts major.

  52. Josef Sanders says

    at 16:20 o’clock it became all clear: since there are no wrong angles in nature there are ipso facto no right angles, only the better angels of our nature. Duh, dudes :)

  53. Ed Seedhouse says

    chrislawson@60: “remember that according to the Heisenberg Uncertainty Principle, we can’t know a particle’s position with perfect precision (as this would imply literally infinite uncertainty in its momentum).”

    We can know a particle’s position to as accurately as we are able to measure, but the more precise our knowledge of position the less we know about it’s momentum. In the limit if we know a particle’s position with perfect accuracy we then have zero knowledge of it’s momentum. But of course we never know anything with perfect accuracy.

    But still the mathematical formality we use for this assumes that a subatomic particle is a mathematical point. That may or may not be “true”, but the mathematical formality provides predictions that are accurate to the limits of our measurements.

  54. Rob Grigjanis says

    Ed @65:

    But still the mathematical formality we use for this assumes that a subatomic particle is a mathematical point.

    The formalisms of standard quantum mechanics (QM) and local quantum field theory (QFT) don’t assume that. They do assume that wave functions (in QM) and fields (in QFT) are local; i.e. that they have values* at each space time point. What both formalisms give you is a way to calculate the probability that a particle will be found in the neighbourhood of a certain point, or within a certain solid angle relative to a scattering point.

    So, in QM, if a single particle wave function is ψ(x,y,z,t), the probability that the particle will be found in a particular volume V at time t is the 3D integral over the volume V;

    ∫|ψ(x,y,z,t)|²dxdydz

    *For a generalized notion of ‘value’. Quantum fields are operator-valued.

  55. consciousness razor says

    Rob Grigjanis:

    So, in QM, if a single particle wave function is ψ(x,y,z,t), the probability that the particle will be found in a particular volume V at time t is the 3D integral over the volume V

    This isn’t meant as a criticism or to start a debate, but to offer some clarification….

    When you say it “will be found” there, you mean more or less that this is what the outcome of your experimental device will indicate (as predicted), after interacting with it in some way. The particle didn’t need to be there before you did the experiment, so that when you came along with your device, you “found” it where it already was.

    The probabilities describe the outcome of that particular kind of interaction, after it occurs, which isn’t literally “finding” it in the normal sense of the word. You could be “putting” it there and calculating the chances of putting it in one place rather than another, not the chances of “finding” it where it had been before the interaction.

  56. Rob Grigjanis says

    cr @67:

    The particle didn’t need to be there before you did the experiment, so that when you came along with your device, you “found” it where it already was.

    I have no idea what you’re talking about. The particle doesn’t “need” to be anywhere. The probability that it is in the specified volume is the integral I gave. That’s just the Born rule. Is the “where it already was” another reference to Bohmian mechanics? If so, you’re interpreting, not clarifying. Note that I specified standard QM.

  57. consciousness razor says

    Is the “where it already was” another reference to Bohmian mechanics? If so, you’re interpreting, not clarifying. Note that I specified standard QM.

    No, it’s a reference to what English-speakers will take to be the ordinary meaning of the word “found,” as we customarily apply it to macroscopic object like me and you, tables and chairs, etc. When you find a chair, it’s understood that it was already in that place wherever you found it. You weren’t producing a location for the chair to have, which it didn’t have prior to your “finding” of it. That’s what finding means. When there are other circumstances which don’t work that way, it’s not going to be understood that you’re “finding” it but as some other kind of circumstance. Perhaps you “put” the chair in that place, for instance.

    If standard QM doesn’t work that way and doesn’t make that kind of distinction (in certain specific situations), because for instance particles aren’t located anywhere unless/until you “find” them, then that doesn’t correspond to the ordinary meaning of the word that people would very naturally assume. So, as I said, it’s not a criticism of QM, I’m not getting into that debate, nobody needs to agree with me about a better version.

    But presumably you would like other people to understand what you’re saying about QM, and all of the care you took to precisely formulate a mathematical expression is pointless, when it’s interspersed with language that ordinary people are bound to misinterpret. Your use of the term in that sentence probably wouldn’t help anyone who doesn’t already understand (much less agree with) the standard/textbook/orthodox interpretation.

  58. Rob Grigjanis says

    Siggy @70: I read your post earlier today in its pristine pre-comment state. That’s good enough!

  59. Friendly says

    All of the interatomic bonds in an octahedral molecule such as sulfur hexafluoride are 90 degrees from the nearest four others.

  60. Matrim says

    There are no right angles, everything is actually on a 1-dimensional line*, we’re just really really bad at experiencing things.

    *the single exception being cats, which exist in a separate intersecting 11-dimensional construct, all dimensions above 5 being some variety of asshole dimension.

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