Plato's Cave - The Orlando Philosophy Meetup Group Message Board › DOES MATH EXIST? (Is math discovered or invented?)

DOES MATH EXIST? (Is math discovered or invented?)

Swami
swami_pixelguy
Orlando, FL
Post #: 491
(Download related file HERE­.)



These are actually age-old philosophical questions. Recently, Plato’s Cave philosophy meetup member, Pamala Clift, posted a video­ that does a great, entertaining job of addressing these questions. Below is a thread­ from a post I made to my ‘WING CIRCLE’ math/philosophy facebook group.

So, check it out. Feel free to add comments to the facebook thread, +/or in this Plato’s Cave discussion post. TIA.

ReallYours,

swami_mathtraveler

P.S. The fb thread includes LOTS of great discussion points and exchanges, plus a little humor along the way:) There are also some nice links worth following. Of particular note is Alan Gaynor’s “On a Notation for Dimensionally Qualified Predicate Calculus Quantification”, including his ‘dimensional cardinality discontinuum’ (of abstract/concrete). He considers combinations from 3 space dimensions and one time dimension. See the nice table in his paper. Also, I posted links to a past Wing Circle meeting addressing ‘HOW BIG IS MATH?’ (overview&thread­, recap­).

Rami K.
Rammy
Orlando, FL
Post #: 589
Being something of a human computer -- by will, and often with flaw, I ask what if math is simply recognized? I mean, are we all not, in some part, pattern matching machines?
Katlaya
Katlaya
Winter Park, FL
Post #: 17
That's the exact video that I told Mike about during our conversation at the meetup. Great video and makes one think.
Ben Forbes G.
Epicurean306
Kissimmee, FL
Post #: 350
Someone can please correct me if I’m naïve or uninformed regarding this question... but I’ve seen no evidence (yet) that math exists independently like Platonic forms or something... Though this idea of “mathematical realism” might not be impossible, I do think it seems improbable. Ergo, I think math is (“merely”?) an elaborate set of internally consistent symbol systems with awesome explanatory power and utility vis-à-vis describing, studying, and even predicting multifarious aspects of empirical ontological reality. I also think that: while there might be common ways (or more or less pragmatic ways) to construct and use various types of math — and math can be “translated” in many cases — there is a sense in which it is patently arbitrary and constructed; for example, pre-Columbian ancient Maya numerals were primarily a vigesimal (base-twenty) system instead of a decimal system (other systems in history have been base-six, etc.), and we can scarcely imagine (and probably have no clue whatsoever) how radically different the mathematics of intelligent extra-terrestrial aliens might be compared our own! Nevertheless, despite some possible (and probably likely) “indeterminacy of translation” — it might be possible to communicate with them through mathematical concepts.biggrin Indeed, it might even be easier than translating each other’s languages (assuming that would even be possible).

Concurrently, mistaking the map for the territory is a little known but very important fallacy where someone may confuse the semantics of a term with what it represents. A similar term is “reification” — where abstractions are taken to be real. Such a fallacy is due to the mistaken belief that a symbol or model is actually the same as the reality that it represents or that one’s measurements are exactly the same as the thing that one measures. The name is a metaphorical representation of mistaking words and symbols for things that they could mean, rather than what they do mean (or those actual things themselves). It is also sometimes referred to as the “Veil Of Māyā” — using the Sanskrit word माया (“māyā”), meaning “illusion” or “delusion” — but Alfred Korzybski referred to it as “the illusion of mistaking the map for the territory” and declared the maxim “the map is not the territory” to summarize this fallacy.

There are also examples of this fallacy in science in the sense of “mistaking the model for the reality” (and particularly in too literal interpretations of scientific theories or interpretations operating under unrealistic biases). This happens when people (sometimes methodologically legitimate but mistaken scientists, and sometimes deluded woo pushers) take a theory, which is no more than a model used to predict nature with as maximal explanatory power as possible, and apply what it says literally.

A sample case would be the principle of least action. Although the equations and theorems to do with this principle are perfectly valid (indeed one can construct them from Newton’s laws) they lead to an “apparent teleology” if taken at face value: that a particle operating under the path of least action must somehow predict its future and move along a path of least action. In reality, such an interpretation is nonsense. In quantum mechanics, particles are often described as “being both a particle and a wave” and this causes confusion and headaches all around — how can something be two totally different things at the same time, exactly? The answer is clearer when thought out in different terms: a quantum particle is actually neither a particle nor a wave, but something completely different that has some properties of both (which are each only meaningful as separate entities operating at our macroscopic-level world anyway, and are thus described in terms and conceptions we’re familiar with as we initially discover and study quantum phenomena).cool

Consider also some of Zeno’s paradoxes (wherein the “maps” seem to fall short of describing the realities of the “territories” involved)
  • Achilles and the tortoise paradox: In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. (as recounted by Aristotle, Physics VI:9, 239b15)
  • Dichotomy paradox: that which is in locomotion must arrive at the half-way stage before it arrives at the goal. (as recounted by Aristotle, Physics VI:9, 239b10)
  • Arrow paradox: if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. (as recounted by Aristotle, Physics VI:9, 239b5)
  • Paradox of Place: if everything that exists has a place, place too will have a place, and so on ad infinitum. (Aristotle, Physics IV:1, 209a25)
  • Paradox of the Grain of Millet: there is no part of the millet that does not make a sound: for there is no reason why any such part should not in any length of time fail to move the air that the whole bushel moves in falling. In fact it does not of itself move even such a quantity of the air as it would move if this part were by itself: for no part even exists otherwise than potentially. (Aristotle, Physics VII:5, 250a20)
  • The Moving Rows (or Stadium): concerning the two rows of bodies, each row being composed of an equal number of bodies of equal size, passing each other on a race-course as they proceed with equal velocity in opposite directions, the one row originally occupying the space between the goal and the middle point of the course and the other that between the middle point and the starting-post. This... involves the conclusion that half a given time is equal to double that time. (Aristotle, Physics VI:9, 239b33)

Rami K.
Rammy
Orlando, FL
Post #: 592

Ben Forbes G.
Epicurean306
Kissimmee, FL
Post #: 354
invisible to humans' natural senses undetectable, or impossible to observe or manipulate by empirical scientific experiment, or metaphysically supernatural, or immaterial, etc...cool

Quantum Entanglement does NOT demonstrate any actual "existence" of these latter sorts of dubious alleged phenomena... and it certainly does not even begin to prove that mathematics have any actual existence outside of their symbolic/conceptual functions when used by sapient brains with awesome explanatory power and utility vis-à-vis describing, studying, and even predicting multifarious aspects of observable, testable, and empirical ontological reality...
Alan G.
Logician7
Orlando, FL
Post #: 27
Since some of us have begun to adopt the notation for dimensionally qualified predicate calculus (thank you!) while others have downplayed its significance, I thought I would write a slightly inflammatory piece (not completely in jest) to spark even further discussion (and hopefully some proofs and conclusions, too). Manifesto of Ontological Agnosticism
John G
user 10908971
Tokyo, JP
Post #: 3
I am enjoying this topic.
John Gomez
christopher
user 11174629
Casselberry, FL
Post #: 128
Since some of us have begun to adopt the notation for dimensionally qualified predicate calculus (thank you!) while others have downplayed its significance, I thought I would write a slightly inflammatory piece (not completely in jest) to spark even further discussion (and hopefully some proofs and conclusions, too). Manifesto of Ontological Agnosticism
Line 4 appears to be a case of question-begging.

Edit -- specifically, the proof purports to show a contradiction in the physicalist claim but does so by assuming physicalism is false to begin with. Clearly if propositions are non-physical then physicalism is false. And if physicalism holds that 'everything is physical' and some proof comes along assuming that 'some things are non-physical' (ie assuming that physicalism is false) then the proof will contradict physicalism. Certainly, obviously.

The physicalist assumption that everything is physical, period, extends to propositions as well, as should be clear from the words 'everything is physical'. Propositions -- mathematical, linguiformal, theoretical, what have you -- are held to be embodied physically in a) the medium in which they are expressed (eg words on a page) and b) the brains in which they are instantiated or realized. The task of the physicalist is to demonstrate this physicality. I should hope we don't need to belabor the point that words on a page are physical. I'm not sure how much belaboring is necessary to show that thoughts are embodied in brains. Let me know.

Edit 2 -- note that my objection here has been made purely from within the context of the argument given in the manifesto without resorting to ad homs or speculating on the psychological motivations (of all things!) underlying said argument. Can't say the same for Alan's original piece. Shame.
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