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In the Wikipedia entry on Philosophy of mathematics, the following is mentioned about Platonism:

[M]athematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging.

What does 'have no causal properties' mean? Does it mean mathematical entities are transdual (beyond duality or binary logic) entities thus any action which rely on relation between cause and effect will not be able to affect them?

Would this also be possible explanation behind 'have no spatiotemporal properties' thing? Because [space|time] is some kind of duality too. So if mathematical entities are transdual they would also automatically outside such constraints.

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    It means that in the sum 2+3=5; two does not act 'causally' on three, and nor are these entities located' spatially' or 'temporally'; we can see this in contrast to two apples+three apples=five apples, here they are 'spatially' and 'temporally' situated (they are there on that table - although we have imagined them), and its my act that 'caused' the two groups to be viewed as one. Sep 27 '17 at 3:27
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    eternal and unchanging are assumptions. they are abstract entities with emphasis on abstract. how can something that is abstract construct of empirical reality transcend duality and cause and effect? Sep 27 '17 at 6:33
  • what does 'have no causal properties' mean? It means that a mathematical "object" (being an abstarc entity) cannot act on some physical object (body, animal, etc.). Sep 27 '17 at 6:59
  • transdual (beyond duality or binary logic) ??? "Mainstream" mathematics assume "binary logic": Excluded Middle and Law of Non Contradiction. Sep 27 '17 at 7:23
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    To go beyond dualism is to go beyond the numbers, so it seems better to say that mathematical entities are part of the created world or 'world of opposites' and do not transcend it. This would be Lao Tsu's view and. as it happens, my own. .
    – user20253
    Sep 27 '17 at 9:07
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There are no considerations of cause and effect or the passage of time in mathematics. (These are in the realm of science.) This is shown most clearly in how logical implications are used in mathematics where P implies Q does not mean that P causes Q, or that Q causes P. It means only that we do not have both P being true and Q being false. We could have both P and Q being true, or both being false. We could even have P being false and Q being true. By saying that P implies Q in mathematics, we are only ruling out that both P is true and Q is false.

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  • Logical implications are used the same way in science, "if X is a rat then X is a mammal" involves no causality. Causality is something additionally stipulated in scientific theories, and one can just as well stipulate causal action of mathematical objects on our "mathematical intuition", as Gödel did. Also, parts of mathematics are constructivist, and there saying that P implies Q does not amount to ruling out P true and Q false, one must give a construction of a proof of Q from a proof of P.
    – Conifold
    Oct 15 '17 at 21:38
  • In science we have causal implications like sub-zero temperatures cause water to freeze. There are no such causal implications in mathematics. And notwithstanding certain fringe elements, the vast majority of mathematicians accept that P implies Q does indeed rule out both P being true and Q being false in proofs. IMHO one is severely limited otherwise and for no good reason. Oct 16 '17 at 1:33
  • Yes, and we have non-causal ones in science as well, as, according to platonists, we have causal ones in mathematics. And I did not mean intuitionists (is that the "fringe elements"?), constructivist mathematics is part of classical mathematics, it is needed in computational applications, etc., and is pretty mainstream. But bare material conditionals are of little use there.
    – Conifold
    Oct 16 '17 at 19:41
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    I understand the computational applications include formal verification that, say, a software implementation written in some programming language will meet the requirements written in some formal specification language. I can see that only a subset of commonly used mathematical techniques might be required in such specialized engineering applications. When I worked as a programmer for several years, I don't think I used floating point arithmetic even once in any of the commercial applications I worked on. I didn't use any material implication either. Oct 17 '17 at 3:51
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"What does 'have no causal properties' mean? Does it mean mathematical entities are transdual (beyond duality or binary logic) entities thus any action which rely on relation between cause and effect will not be able to affect them?"

Primarily it means, in Plato, they aren't part of the things that come into being, and change. They are perpetual. Not part of time. For instance, if one asks, where is 3? One does not need to wait, for 1, 2, and then, only then, 3. It's already, and always, before 4, and after 2. It seems, in the Platonic discussions, that some faculty of the mind must be able to grasp the region where stuff that does not have to rely on coming to be, here or there, this year or that year, always are. The same thing, exactly the same, is held of the human being as a form, i.e., as something that always can be. But, in actuality might not be. In other words, there might be no human beings in the world, or anywhere, but, according to this way of thinking, there always could be, and why? Because of the eternal form or idea. It's like saying that the possibility is real, and doesn't depend on the history or development of things. (Though, one should add, one can't give a definitive answer to how Plato thought about this, but can only derive some degraded sense through studying his, and other ancient works.)

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"Mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging." This sentence is pure religion or "matheology". Everything that can interact with part of reality (like a brain or a computer memory) has spatio-temporal properties. This is true for every philosophical variety including Platonism.

A simple example: The number 3 has lots of representations, for instance one on this screen, another one in the holy trinity. (Without any representation the number 3 could not appear in mathematical discourse and therefore would not exist in mathematics as far as mathematics is accessible to mathematicians.) Every representation is therefore an important part of this number 3. We can say the number consists of a value and its representations, some of which may be as feeble as chemo-electrical processes in a brain. And every representations covers spatio-temporal co-ordinates.

Further every variable is a mathematical entity. As its name says, a variable can vary in value or other properties. A math book is certainly a mathematical entity. It can vary by writing notes into the book and by having new editions. A rope or its shape is a mathematical entity. Both can vary. The representation of a number is a mathematical entity. It can vary according to the system where it is expressed.

With respect to eternity, there are differing opinions. Those Platonists who believe with Cantor that God has created the numbers will insist that they are eternal. There are others however, who accept Dedekind's standpoint: "Every time when there is a cut (A1, A2) which is not created by a rational number, we create a new, an irrational number a which we consider to be completely defined by this cut (A1, A2)". They will not believe in eternal numbers, because "we" can't have created numbers more than some millenia ago.

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    We can "interact" with centaurs and unicorns, but they are not usually considered "part of reality". It makes sense to say that mathematical entities are simply useful fictions, but I see none in saying that representation of something is its part (my portrait is not a part of me) or that mathematical book is a mathematical entity (book about food is not a food entity). In this post it is even hard to understand what position you are trying to describe as an alternative to platonism. Fictionalism? Structuralism? Constructivism? Aristotelian realism?
    – Conifold
    Sep 27 '17 at 23:46
  • Conifold, you cannot interact with centaurs and unicorns - only with your ideas of them. These ideas, illusions, and pictures are part of reality. Books about food are not food. Food is something that exists without our knowledge. Mathematics is nothing without the knowledge about it. The position that I describe is called MatheRealism.
    – Heinrich
    Sep 28 '17 at 10:11
  • Conifold, what you write here is part of my picture of Conifold. Since I don't know anyhing else of you, it is an essential part of your personality for me. Everybody who sees a portrait of you will connect it with your person. I think it is too restrictive to believe that a person ends with its skin.
    – Heinrich
    Sep 28 '17 at 10:16
  • If you consider representations to be part of the thing then our ability to interact with pictures of unicorns counts as interacting with unicorns. If not, interacting with 3 marbles does not count as interacting with 3 either. Food is a relational abstraction, we designate something as food, without us instances of meat and vegetables aren't "food", they just are. So are numbers like unicorns, like food, like me, or yet something else?
    – Conifold
    Sep 29 '17 at 0:25
  • Conifold, you are right of course. As I see it, there are at least three kinds of existence: (a) Material things with or without labels or names in memories, (b) ideas with representations like "electron" or "number three", (3) ideas with only representations like unicorns. But as I write this I see several problems. What exactly is the difference between "electron", "number three" and "unicorn"? There are pictures of electrons and unicorns. No unicorns, but parts of unicorns, for instance organic molecules. I think this question deserves to become a real philosophical question.
    – Heinrich
    Sep 29 '17 at 6:40
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Wikipedia entries are easily edited and I would suggest to remove the 'spatiotemporal'. It is conceivable that mathematical entities are ontologically different but they share some properties with other domains of existence. Actually this explains why geometry works in physics and everyday life: we can calculate how much glass we need for a square window. And a map, which is 2D, is useful even if the world is 3D and there is no good 'mapping' (unambigous and continuous) between such spaces.

Geometrical squares are more durable than square windows but that is not an objection to sharing properties. Encoding time as a line transformed geometry (historically, that is ) into physics:: irreversibilty was left out but some perennial features of pendulums and the like were seen.

The notion of 'cause' is rather anthropomorphic and is used in science mostly as a way of speaking. It can be reformulated with more strict logical means. A good point to remember is that much of logic admits graphical representation (Venn diagrams, etc), so logic does not unconditionally 'transcend' the world.

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