[HN Gopher] How the modern world arose from imaginary numbers
___________________________________________________________________
How the modern world arose from imaginary numbers
Author : CapitalistCartr
Score : 23 points
Date : 2022-02-10 12:13 UTC (10 hours ago)
(HTM) web link (nautil.us)
(TXT) w3m dump (nautil.us)
| adityapatadia wrote:
| This is one video which shows history of imaginary numbers and
| their usefulness in current times.
|
| It's from Veritasium: https://www.youtube.com/watch?v=cUzklzVXJwo
| gumby wrote:
| "Imaginary" was an unfortunate choice of word. Positive integers
| and rationals feel "legit"* to most people, even (or especially)
| the non-mathematically trained.
|
| But "imaginary" numbers are no more or less legit than, say,
| transcendental numbers. But in middle school you learn this
| confusing thing and maths starts to fall off the rails.
|
| * Had to find a synonym for "real" here.
| jdrc wrote:
| Yeah we should stop using the word imaginary, and just call
| them iNumbers (sorry Apple)
| Qem wrote:
| I like to call them "double-barreled numbers". I never liked
| the term "imaginary", because that gives mystical
| connotations and even hamper understanding, in my opinion.
| intrepidhero wrote:
| That's what I've finally settled on in my mind. Since we use
| imaginary numbers to model and measure real phenomenon, they're
| not imaginary in the sense that I would usually use that word.
| There's got to be a better name... Transcendental complement
| maybe.
| viktorcode wrote:
| I highly recommend video by Sabine Hossenfelder [1] on this
| topic. She's explaining the topic in layman terms, starting from
| what does it mean to exist (or be real) in physical sense.
|
| Complex numbers, as useful as they are, just an abstraction, a
| tool which can be replaced with different forms of calculations.
| That makes the question of "reality" rather complex (pardon the
| pun).
|
| [1] https://www.youtube.com/watch?v=ALc8CBYOfkw
| Psyladine wrote:
| For those interested I also highly recommend Veritasium's
| video[1] as an excellent explainer as well, especially
| regarding the roots of imaginary/complex numbers in geometry.
|
| [1] https://www.youtube.com/watch?v=cUzklzVXJwo
| [deleted]
| nuncanada wrote:
| Mathematicians have been constantly been giving bad names do
| their concepts... Imaginary Numbers is one of the atrocious ones,
| Imaginary Numbers are part of the fabric of our reality...
| Meanwhile Real Number were axiomatized in a way that creates
| completely unrealistic numbers, "ghost-like" numbers, numbers
| that can never be given a name... The first course in Analysis is
| ridden with examples of Continuous Functions that are not
| continuous in any sense humans would consider ever it...
| dandanua wrote:
| Life is complex - it has real and imaginary parts.
| 1970-01-01 wrote:
| I've been calling them invisible numbers most of my life. It
| makes much more sense.
| Agamus wrote:
| I'm not an expert here, but truly interested to hear responses to
| this question.
|
| To say that 1+1=2 is "true", does that not require a corollary in
| "reality" to something fundamental that can be called a "one"
| object? I believe this is called mathematical constructivism.
|
| Imagine, hypothetically, that we cannot identify something that
| is physically fundamental and individual. My question is whether
| any mathematics in that scenario could be considered "true"
| without such constructivism, in other words, without a physical
| correspondence to an unquestionably, physically fundamental "one"
| object.
| hgomersall wrote:
| There is a persuasive school of thought that argues exactly the
| opposite, that imaginary numbers are not real
| http://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-are... .
| topaz0 wrote:
| I'm familiar with (and partial to) the geometric algebra
| literature, but I think it's a bit disingenuous/clickbaity to
| say that it implies imaginary numbers aren't real. I think of
| it as meaning that the algebra of the imaginary numbers arises
| as a slice through a more useful, bigger algebra.
| hgomersall wrote:
| Yes, I think it's rather tongue in cheek. It's probably not
| defined as clickbait since it was written in '93!
| dboreham wrote:
| This is only mentioned in passing in the article, but the penny
| dropped for me when I read somewhere else that you should first
| think about negative numbers. They seem pretty real, but they're
| not. There's no negative quantity in nature. You can't have -5
| rocks. Negative numbers are just the solution to equations of the
| form x + n = 0. Now after you let that sink in, imaginary numbers
| don't seem so weird after all. Just another kind of number
| defined as the solution to some equation.
| TheOtherHobbes wrote:
| You can owe someone five rocks (or dollars, to make it only
| slightly less abstract.)
|
| You can go five miles east instead of west (more of a vector,
| but still makes sense if your movements are limited to a number
| line instead of a number plane).
|
| Imaginary numbers are more of the same with a twist. i is just
| a pi/2 rotation on a plane instead of a negative number, which
| can be thought of as rotation by pi on a line.
|
| The big mistake with imaginary numbers is calling them
| imaginary. There's nothing imaginary about them. They're a very
| specific kind of operation which can be expanded with very
| little thought or effort to complex numbers, which have
| incredibly useful properties in engineering.
|
| Calling them "imaginary" is cripplingly confusing for almost
| everyone, and many never get over it.
| redler wrote:
| You could have five exactly rock-shaped holes in the ground,
| and by adding seven rocks, only two remain above the surface.
| mettamage wrote:
| I disagree.
|
| > You can't have -5 rocks
|
| You can have 5 rocks destroyed in the future. That to me, is
| what -5 physically means: a guarantee that the object
| associated with the number will be eliminated out of existence
| in the future and decrement the negative number by 1.
|
| Some people call it debt, but to me, emotionally that word
| feels too financial. So I prefer "a guarantee to be eliminated
| out of existence in the future ", or something like that.
|
| Conversely, 5 cows means: 5 cows currently in existence.
|
| 5 cows - 5 cows means: I see 5 cows and now they don't exist
| any more and there is nothing.
|
| I wish my math skills were better, I am optimistic that I'd
| find a similar thing for imaginary numbers and maybe even
| complex numbers.
|
| With that said, I do get where you're coming from and I find it
| a compelling perspective as well. It's simply that I feel the
| perspective I described as well.
| easywood wrote:
| >> You can have 5 rocks destroyed in the future You have 5
| rocks now, and zero rocks in the future. At no point are
| there negative rock-shaped holes in the fabric of space. The
| parent's comment still stands, he is talking about reality,
| not our mathematical interpretation of it.
| marcus_cemes wrote:
| > You can have 5 rocks destroyed in the future.
|
| Exactly, you use it to store some information that has no
| real quantity but may be converted to a real quantity in the
| future through some other process.
|
| I'm a polytechnic university student, we use imaginary
| numbers extensively in all sorts of places, especially
| whenever there is any oscillatory behaviour, such as an
| electrical signal or a light wave. A complex number is just a
| two-dimensional vector with real/imaginary components, whcih
| provides an amplitude and a phase (angle). An oscillating
| sinusoidal signal/wave may appear to be zero and completely
| static if you freeze time at the right moment, but as time
| progresses, it will continue oscillating, like a swing in a
| park.
|
| In a way, the magnitude represents the built up "momentum" of
| the system, whilst the real quantity is the immediate
| physical value at any given point in time (given by the
| phase). The amplitude is always the same at any given moment,
| even when the swing is vertical, it has momentum which will
| help it reach its maximum height.
|
| Personally, I still think they are just "invented", but I
| think the vast majority of engineers much prefer them to the
| alternative, manipulating trigonometric functions (every
| engineer's nightmare). They're a neat way to represent the
| exchange of potential and mechanical/electrical energy with a
| single value and some simplified mathematics (this is an
| engineer's, not a mathematician's, point of view). Like
| negative numbers, we could have chosen to have two positive
| quantities, balance and debt, instead we find use in merging
| these definitions, whether negative values make sense or not.
| We have become used to to negative numbers representing the
| "inverse" action, which makes sense when representing a
| phyiscal quantity such as velocity.
| taco_emoji wrote:
| You're not actually disagreeing. OP is saying the negative
| numbers are useful _conceptually_ , which is what you're
| demonstrating here.
|
| Also thinking about a number line is useful when talking
| about both negative numbers AND complex ones: negative
| numbers are to the left of 0, but complex numbers are _up and
| down_ from the number line.
| phendrenad2 wrote:
| Imaginary numbers are just a shorthand way of representing things
| in nature that relate to one another via the sine function, for
| example, charged particles in a magnetic field. You can bust out
| calculus to describe the motion, or you can use a convenient set
| of rules that represents the partially-solved equation.
| nyc111 wrote:
| > For some reason--whether a sense that there was some mistake,
| or someone copied something down wrong, or because it was so
| absurd--the manuscripts we have show that Heron ignored the minus
| sign and gave the answer as [?]63 instead.
|
| I wonder if the minus sign was in use in the time of Heron (First
| century AD). I couldn't tell from this Wikipedia page
| https://en.wikipedia.org/wiki/Plus_and_minus_signs
| dTal wrote:
| Another way to think of it is that _all_ numbers are imaginary.
|
| Numbers aren't real. Platonism is wrong. Imaginary numbers aren't
| "out there" somewhere. The whole system of mathematics is an
| accumulated edifice of metaphors designed by human brains, for
| human brains, and there's no god "behind the curtain". It's just
| a tool of thought. It reflects the "reality" of the universe only
| insofar as we've looked at the universe, noticed patterns, and
| constructed metaphors around them.
|
| This is not a popular viewpoint! But it is the only
| scientifically supported one.
| voldacar wrote:
| This is one particular point of view, and it's extremely sloppy
| reasoning to say that it's "scientifically supported", given
| that mathematics is not a form of science.
|
| There are plenty of very smart people, not just mathematicians
| but also physicists & scientists who are mathematical
| platonists.
| dTal wrote:
| There are plenty of smart people - scientists even - who
| believe in all kinds of deities.
|
| Mathematics is indeed not a form of science. But the
| existence and shape of mathematics is an observable
| phenomenon, and so _meta_ mathematics - the study of what it
| is and where it comes from - _can_ be studied scientifically.
| How do you know mathematics exists? Well, there 's a textbook
| right there. Who wrote the textbook and why? A human,
| expressing metaphors inside their heads. How did those
| metaphors get inside that human's head? Ah, well, that's the
| interesting bit - the answer of course transpires to be "a
| combination of innate ideas imprinted by genetic evolution by
| natural selection, and sociology". And you don't have to stop
| there, you can explore in glorious detail exactly _where_
| each idea comes from, what innate monkey-ish tendency is
| being deployed, how exactly ideas like "infinity" fit in a
| mind designed for finding fruit and chasing things.
|
| We can similarly bring all manner of religious beliefs under
| the anthropological knife. It's not a pretty process though,
| to the people who believe in them.
| carapace wrote:
| > metaphors inside their heads.
|
| You're begging the question presupposing the non-Platonic
| viewpoint here. How do we know that metaphors are "inside
| the head"?
| voldacar wrote:
| You are just assuming the point you're trying to prove
|
| > There are plenty of smart people - scientists even - who
| believe in all kinds of deities.
|
| Okay? This is supposed to make me feel - how exactly? I'm
| not inherently disdainful towards theism or theists, but if
| I were, I guess your remark would make me like science
| less, or something?
|
| > We can similarly bring all manner of religious beliefs
| under the anthropological knife
|
| I'm not really sure we can, actually. At least not in some
| kind of non-contentious, "objective" sense. I don't really
| trust individual humans to give an accurate account of why
| they believe their beliefs, but I trust "anthropology" and
| "sociology" even less. My distrust for this on an
| individual scale comes from the fact that many beliefs &
| memes exist for purposes of social signalling, group
| identification, etc, and it might not actually be in your
| interest to know exactly why you believe what you do.
|
| But these auxiliary functions of beliefs, such as
| signalling etc, seem to me to scale _up_ as you introduce
| groups and larger-scale activities such as "anthropology"
| and "sociology". Without some feedback loop keeping them
| honest, why would I expect anthropologists or sociologists
| to tell me a true story about why someone believes what
| they do, any more than that person or anyone else? In
| aerospace engineering, the feedback loop is that if your
| design is bad, your jet engine won't work. As a result, I
| generally trust aerospace engineers about jet engines. But
| what is there to stop sociologists, anthropologists, etc
| from just settling on some bullshit that agrees with their
| preconceived beliefs or flatters their group status and
| promoting it forever?
|
| But back to math. The history of mathematical ideas is
| complicated and interesting, but it isn't really that
| relevant to the question of whether the things those ideas
| are _about_ are "real", which is equivalent to asking
| whether mathematical platonism is true or not. The question
| of platonism comes down to the definition of words like
| "real" and "exist". It is very easy to equivocate using
| these words, which is why most discussions about
| mathematical platonism are so low quality. I think the
| overall question isn't that meaningful so I'm not really a
| platonist or an anti-platonist. In most parts of human
| life, when I say "x exists", I mean that I can reach out
| and touch x, that it has a mass, temperature, surface
| texture, etc. In math, when I say "x exists" I just mean
| that I can talk about x without creating any logical
| contradictions. The square root of -1 may not exist in the
| same sense as my laptop here, but it exists in the sense
| that I can do things with it, such as add, multiply, raise
| to powers, etc, without reaching a contradiction in my
| formal system. So the whole "out there" thing doesn't
| really matter. There doesn't need to be an "out there" in
| order for me to meaningfully say that the square root of -1
| exists.
|
| I think that a lot of philosophy is like this too, when you
| mentally zoom in really closely on a problem, it often
| reduces to some kind of equivocation or inconsistent
| language usage.
|
| Btw I don't really consider anthropology or sociology to be
| real intellectual disciplines, and I'm pretty on the fence
| about psychology and economics. I realize that is an
| unpopular opinion but I've thought about it a lot and I'm
| pretty certain that it's correct. Aerospace engineering is
| real because it _attaches_ to some fundamental reality,
| namely that of the spinning fan blades, the combusting
| fuel, etc. If you get your engineering wrong, the fan
| blades won 't spin. Likewise, math is attached to systems
| of axioms. When your do your math wrong, you get a
| contradiction. Sociology and anthropology don't attach to
| anything, they're like a closed loop, like theology. If you
| get your anthropology wrong, nothing really happens.
| nathias wrote:
| No it isn't scientifically supported because science does not
| include ontology and episemology as its domain. I'm not a
| platonist, but the reasons for not being platonists are
| philosophical.
| dandanua wrote:
| > there's no god "behind the curtain"
|
| have you looked there already? :)
| mjburgess wrote:
| This view isnt "scientifically supported" because science is
| neutral on (indeed, even presupposes) the existence of abstract
| objects.
|
| No one believes abstracta have a physical location -- they lack
| physical properties. The claim "2 + 2 = 4" is true -- and
| clearly not true invirute of anything anyone thinks... if we
| kill that person (/people), it is no less true.
|
| Indeed, if numbers don't exist (for example), do we suppose
| that we can't communicate issues of quantity with other species
| (, & possible alien life, etc. etc.) ? (If we can, what shared
| things are we talking about when we quantify?)
|
| It seems deeply implausible to say that our use of number is
| circumstantially psychological -- any description of reality is
| going to be indispensably quantitative --- quantity _is_ what
| we are talkng _about_. We are not talking about ourselves.
| echopurity wrote:
| jhedwards wrote:
| I've thought about this problem quite a bit and, while my
| initial position was the same as above (math is not "real"
| per se) I had to concede that integers are real, because
| quantity is self-evidently real.
|
| If you have four oranges, the quantity "four" is right there.
| If you take away one of those oranges you know that the
| result cannot be split evenly without a remaining orange
| because of the properties of odd numbers.
|
| If you cut the remaining orange in half then you get a
| rational number, but is that self-evidently real? The halves
| of the orange are only "halves" because we consider them in
| relation to their origin, which we consider to be "one"
| orange. So rational numbers necessarily involve the human
| action of relating some quantity to a reference quantity,
| therefore they are a higher-level abstraction built on top of
| the fundamental physical property of quantity.
|
| In the end I decided that math is based on a foundation of
| quantity (and maybe "space" as well?) and everything else was
| a derived abstraction. I am very curious if anyone else has a
| good argument for other parts of math being fundamental.
| viovanov wrote:
| Does four really exist as 4? Maybe it's just 2 squared.
| dagw wrote:
| _integers are real, because quantity is self-evidently
| real._
|
| But the there are more integers than there are quantifiable
| 'things'[1]. Are integers that are a lot larger than, say
| the size of the power set of all fundamental particles in
| the universe still "self-evidently real".
|
| [1] Assuming a finite universe (or a finite number of
| finite universes) and a few other things.
| ajuc wrote:
| Electromagnetic field changes are described by complex
| numbers. So not only you need fractions, you need
| irrational numbers and imaginary numbers to describe the
| universe. Why is counting oranges "self-evidently real" and
| describing electrons "kinda real"?
|
| I'd argue the opposite - oranges never appear in the laws
| of physics. They are just our description of a collection
| of atoms sharing some pretty loosely-defined
| characteristic. Oranges aren't perfectly equivalent to each
| another, so whether you count 1 small and 1 big orange as 2
| or 1.5 oranges depends on your arbitrary decision. How
| about 1 orange and 1 hybrid species between orange and
| grapefruit? How close you need to be to be considered
| orange? Classes of equivalence are determined by us not by
| the universe, and numbers are derived from that.
|
| Electrons on the other hand are as undeniably real as
| anything in this universe can be.
| jhedwards wrote:
| Quantity has real concrete measurable effects that exist
| irrespective of the philosophical problem of
| classification. If I have two acorns I know I can
| potentially grow two very real trees. They are countable
| and that directly relates to the effect they can have on
| the world. I like to think that maybe every tree is one
| tree, or that all trees are part of a unity of "plants",
| but practically speaking seeds and trees are countable
| entities no matter how I classify them.
|
| If there are two planets, we can discuss philosophically
| that one might be a "moon" and not a "planet", or in some
| sense that the planet is "continuous" with the space dust
| or whatever. But the existence of two distinct bodies in
| space will still create very specific gravitational
| fields from their interactions. Tides are different if
| you have one vs two moon, Lagrange points etc.
|
| As for electromagnetic fields, I am not smart enough to
| make a judgement on that. They are described by complex
| numbers, but does that mean they reflect a physical
| embodiment of complex numbers? Or is it just that we
| require complex numbers in order to resolve their
| behavior into something measurable? I love to learn about
| electricity but sadly the math is beyond my ability.
| pdonis wrote:
| _> Electromagnetic field changes are described by complex
| numbers._
|
| You can do this, but there's no need to. You can describe
| electromagnetism using only real numbers.
|
| A better argument for imaginary numbers being necessary
| to describe the universe is quantum mechanics, since
| quantum interference (in particular destructive
| interference) means that two possible events that each
| have a positive probability taken in isolation can cancel
| each other out, implying that probabilities can combine
| with a minus sign. And that means that probability
| amplitudes, which are square roots of probabilities, can
| have nonzero imaginary parts.
| mettamage wrote:
| 4 oranges are real because we have the neural architecture
| to classify the oranges as belonging to the same group
| according to whatever our classification criteria are.
|
| What if you can't classify but only be conscious of input?
| Kinda like being in a super dreamy state (or psychedelic
| one). From that state of consciousness, numbers aren't real
| but reality can be (in the psychedelic case).
|
| Just brainstorming
| darkscape wrote:
| > I am very curious if anyone else has a good argument for
| other parts of math being fundamental.
|
| Groups. You can stay in your kitchen (the neutral element)
| or go into the bedroom, then come back (inverses). In my
| mind, this is as real as quantity.
| hansbo wrote:
| But even if you deep dive enough, there are discrete values,
| like in Quantum Mechanics. And as long as you have discrete
| values, you have integers, no? So integers do not seem only
| like human models, they seem to me as something innate in the
| universe.
| helen___keller wrote:
| Quantum mechanics is a mathematical description of the
| behavior of the universe, so wouldn't invoking this to prove
| mathematical objects exist be begging the question?
|
| Not to say I agree with GP, but I don't think it will be so
| easy to prove GP wrong either
| dTal wrote:
| "Discrete values" are also a human metaphor. You say there
| are two apples on your desk? I say there is a fuzzy quantum
| mess of probability distribution functions on your desk. "Two
| apples" is in your mind.
| nh23423fefe wrote:
| and spin?
| shusaku wrote:
| I'm baffled that you're invoking QUANTum mechanics to
| ascertain that discrete values don't exist. At any rate,
| nominalism has a rich history, so I doubt these hacker news
| comments will solve the issue...
| igorkraw wrote:
| are atoms discrete? we used to think so. we might never get a
| better model than quantum physics and it might still be wrong
| and fail to explain things. So there is a human idea of
| "discrete element" that we used to apply to everything - and
| as we look closer, it always breaks down. that doesn't mean
| it's a useless abstraction, but it is am abstraction - a tool
| for thought, a map, not the territory
| andreareina wrote:
| Science has nothing to do with it, rather it's a question of
| philosophy and what we define as being real.
| dTal wrote:
| Science is "what we have evidence for". Is there evidence for
| some abstract mathematics that we didn't invent?
| throwaway17_17 wrote:
| I agree that this view it is not popular, but I also do not
| think that supporters often articulate their view/support well.
| I am a hard materialist and the amount of platonic-leaning
| discourse around the fundamentals of mathematics confuses me. I
| do not know how so many people (typical those outside
| philosophy and mathematical foundations) just assume a platonic
| style view.
| dTal wrote:
| I am currently reading "Where Mathematics Comes From" by
| George Lakoff and Rafael E. Nunez - the same Lakoff who
| authored the seminal "Metaphors We Live By", so I have a lot
| of time for him. At first it seems like they're just going to
| explore the pedagogical psychology of mathematics - how
| interesting! But then right at the end of the preface they
| hit you with "and by the way this is all there is to it,
| mathematical Platonism is a lie", which struck me immediately
| as straying out of their lane. But it seems their
| investigation into the titular question overwhelmingly led
| them to this conclusion. The argument is pretty simple - if
| there is a "platonic mathematics", we cannot have any direct
| experience of it. All mathematical thought, like all thought
| in general, is metaphorical. The predictive power of
| mathematics in the real world is unsurprising because we
| throw away the metaphors that don't work well.
|
| I do not _like_ this conclusion. Mathematics has always been
| something of a religion for me. But I can find no flaw with
| the argument. From a scientific perspective, mathematics
| bottoms out at "what goes on in human noggins".
| imbnwa wrote:
| >The argument is pretty simple - if there is a "platonic
| mathematics", we cannot have any direct experience of it.
|
| Aside, but this is also Aristotle's exact argument against
| Platonism in general, though when he makes it in the
| Nichomachean Ethics he is specifically talking about
| ethical Good (if the definition/actual taking place of the
| Good lies in some other plane, we can't participate in it
| so no one is or can be good), but the idea is the same even
| when he's talking about what a soul is in De Anima.
| Aristotle doesn't believe in 'souls' in the way we think of
| them as religio-spiritual entities that exceed the capacity
| of the body; a 'soul' for Aristotle _is_ the body but in a
| way that radically challenges the idea of a body as mere
| shell or vessel - soul is what any form of life repeats
| doing, as a body, in order to continue being itself. It
| should be noted that a lot of time at Aristotle 's Academy
| was spent in Zoology, studying animals and their anatomy.
| darkscape wrote:
| I'll have to read the book, but in my mind, the (emprical)
| study of humans and their brains doesn't shed light on the
| metaphysical question of the nature of mathematics. What
| they find is how humans have developed to do mathematics.
| We could have evolved to be the way we are with or without
| mathematics being "out there". Survival in the physical
| world would lead us to "throw away the metaphors that don't
| work well". At any point in time a concrete human being
| would still be able to consider only a limited set of
| mathematical ideas i.e. for humans "mathematics bottoms out
| at "what goes on in human noggins"".
|
| I'd say the patterns you mentioned in an earlier comment
| are a way for math (or parts of it e.g. some integers) to
| be "out there". If humans embody mathematics, then
| analogously so do those patterns.
| mannykannot wrote:
| I don't think the issue has been as definitively settled as
| you have been persuaded to think. Let's take a look at the
| claim "if there is a 'platonic mathematics', we cannot have
| any direct experience of it" (I realize this is probably a
| paraphrase of a fuller argument, but it is what I have to
| work with here.)
|
| Firstly, note the word "direct" here. If it has any
| relevance, then the authors have assumed the burden of
| explaining either that there are only direct experiences,
| or why indirect experiences don't count.
|
| Secondly, what are the premises here? If this is supposed
| to be axiomatic, then there is literally no reason to
| either accept or reject it, and claims that the issue has
| been settled are just statements of belief; otherwise, the
| argument needs to have premises that are not begging the
| question in some way. As it stands, this claim is not an
| argument; it is more of an intuition pump.
|
| Metaphysical discussions tend to (always?) end up as being
| about the meaning of words like 'real' and 'true'. Whether
| such discussions can really tell us anything about what
| must be true is arguably the most meta question in
| metaphysics.
| tim333 wrote:
| >all numbers are imaginary.... mathematics is an accumulated
| edifice of metaphors designed by human brains
|
| Or you can say they exist but in a different way to physical
| reality.
|
| I mean pi probably still was 3.14159... before humans evolved
| so it's not our fault really.
|
| Personally I think maths not only exists but physical reality
| is a subset. I mean why else is there something rather than
| nothing? Scientifically it's the only hypothesis that works for
| that really.
| mannykannot wrote:
| > But it is the only scientifically supported one.
|
| Really? What is the empirical evidence for it?
| canjobear wrote:
| Reality is that which, when you stop believing in it, doesn't
| go away. -Philip K Dick
|
| Even if you stop "believing in" math, your proofs are still
| either correct or incorrect.
| dagw wrote:
| The proofs are only correct in as far as you believe in the
| Axioms of mathematics that those proofs are built on. Stop
| "believing in" the axioms and the proofs are no longer true
| or even meaningful.
| canjobear wrote:
| But it is still the case that IF the axioms hold, THEN the
| proofs are either valid or not.
| marcosdumay wrote:
| So it's conditional reality. Now make that fit within
| your usual realistic philosophy...
| canjobear wrote:
| Easy. There exist contingency relationships between
| axioms and theorems provable from those axioms.
| marcosdumay wrote:
| Oh, yeas, your proof is quite real. And it's completely
| meaningless, being all about imaginary things.
|
| It can only have any meaning if you adhere to some
| scientific model.
|
| At this point you've deviated so much from the OP
| discussion that you could as well talk about angels
| dancing in pinheads. Any quantification of them is as
| real as your proof.
| tasha0663 wrote:
| You can boil down pretty much everything to "an accumulated
| edifice of metaphors designed by human brains, for human
| brains".
|
| In the game Hearts, if you take most of the spades you lose.
| However, if you manage to take _all the spades_ you win, and
| they call it "shooting the moon".
|
| In a similar fashion, when you reject everything as an unreal
| system of metaphors, Platonism "shoots the moon" by having us
| reexamine what we thought we meant by "real" in the first
| place.
| rthomas6 wrote:
| What finally made imaginary numbers intuitively make sense to me
| is realizing that the number i just represents a 90 degree
| rotation on the complex plane. That's why i*i = -1 (it's rotating
| 180 degrees), why imaginary numbers are orthogonal to real
| numbers, why e^ix = cos(x) + i*sin(x) makes sense, everything.
|
| When you are dealing with 2 dimensions, complex numbers are a
| kind of hack for representing both dimensions without any kind of
| vectors or pairs or anything, just numbers.
| polotics wrote:
| Mathematicians have always sucked at naming their variables, and
| constants, and types... This has given pundits of all ilk
| countless opportunity to debate...
___________________________________________________________________
(page generated 2022-02-10 23:00 UTC)