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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism

[reasons for believing maths entities exists]

34 ideas
One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato]
     Full Idea: If one is, there must also necessarily be number - Necessarily - But if there is number, there would be many, and an unlimited multitude of beings. ..So if all partakes of being, each part of number would also partake of it.
     From: Plato (Parmenides [c.364 BCE], 144a)
     A reaction: This seems to commit to numbers having being, then to too many numbers, and hence to too much being - but without backing down and wondering whether numbers had being after all. Aristotle disagreed.
We aim for elevated discussion of pure numbers, not attaching them to physical objects [Plato]
     Full Idea: Our discussion of numbers leads the soul forcibly upward and compels it to discuss the numbers themselves, never permitting anyone to propose for discussion numbers attached to visible or tangible bodies.
     From: Plato (The Republic [c.374 BCE], 525d)
     A reaction: This strikes me as very important, because it shows that the platonist view of numbers places little or no importance on counting, inviting the question of whether they could be understood in complete ignorance of the process of counting.
In pure numbers, all ones are equal, with no internal parts [Plato]
     Full Idea: With those numbers that can be grasped only in thought, ..each one is equal to every other, without the least difference and containing no internal parts.
     From: Plato (The Republic [c.374 BCE], 526a)
     A reaction: [Two voices in the conversation are elided] Intriguing and tantalising. Does 13 have internal parts, in the platonist view? If so, is it more than the sum of its parts? Is Plato committed to numbers being built from indistinguishable abstract units/
Geometry is not an activity, but the study of unchanging knowledge [Plato]
     Full Idea: Geometers talk as if they were actually doing something, and the point of their theorems is to have some effect (like 'squaring'). ...But the sole purpose is knowledge, of things which exist forever, not coming into existence and passing away.
     From: Plato (The Republic [c.374 BCE], 527a)
     A reaction: Modern Constructivism defends the view which Plato is attacking. The existence of real infinities can be doubted simply because we have not got enough time to construct them.
We master arithmetic by knowing all the numbers in our soul [Plato]
     Full Idea: It must surely be true that a man who has completely mastered arithmetic knows all numbers? Because there are pieces of knowledge covering all numbers in his soul.
     From: Plato (Theaetetus [c.368 BCE], 198b)
     A reaction: This clearly views numbers as objects. Expectation of knowing them all is a bit startling! They also appear to be innate in us, and hence they appear to be Forms. See Aristotle's comment in Idea 645.
It is a simple truth that the objects of mathematics have being, of some sort [Aristotle]
     Full Idea: Since there are not only separable things but also inseparable things (such as, for instance, things which are moving), it is also true to say simpliciter that the objects of mathematic have being and that they are of such a sort as is claimed.
     From: Aristotle (Metaphysics [c.324 BCE], 1077b31)
     A reaction: This is almost Aristotle's only discussion of whether mathematical entities exist. They seem to have an 'inseparable' existence (the way properties do), but he evidently regards a denial of their existence (Field-style) as daft.
Numbers seem to be objects because they exactly fit the inference patterns for identities [Frege]
     Full Idea: The most important consideration for numbers being objects is that they sustain the patterns of inference demanded by the reflexivity, transitivity and symmetry of identity.
     From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]), quoted by Crispin Wright - Frege's Concept of Numbers as Objects 1.iii
     A reaction: But then if I say that the 'whereabouts of Jack' is identical to the 'whereabouts of Jill', that would seem to make whereaboutses into objects.
Frege's platonism proposes that objects are what singular terms refer to [Frege, by Wright,C]
     Full Idea: The basis of Frege's platonism is the thesis that objects are what singular terms, in the ordinary intuitive sense of 'singular term', refer to.
     From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Crispin Wright - Frege's Concept of Numbers as Objects 1.iii
     A reaction: This claim strikes me as very bizarre, and is at the root of all the daft aspects of twentieth century linguistic philosophy. See Bob Hale on singular terms, who defends the Fregean view against obvious problems like 'for THE SAKE of the children'.
How can numbers be external (one pair of boots is two boots), or subjective (and so relative)? [Frege, by Weiner]
     Full Idea: If the number one is a property of external things, how can one pair of boots be the same as two boots? ...but if the number one is subjective, then the number a thing has for me need not be the same number the object has for you.
     From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Joan Weiner - Frege Ch.4
     A reaction: This nicely captures the initial dilemma over the nature of numbers. It is the commonest dilemma in all of philosophy, struggling between subjective and objective accounts of things. Hence Putnam's nice definition of philosophy (Idea 2352).
Identities refer to objects, so numbers must be objects [Frege, by Weiner]
     Full Idea: Identity statements are about objects. If we can say that 1 is identical (or not) to 0, then 1 must be an object.
     From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Joan Weiner - Frege Ch.4
     A reaction: This seems to point to Platonism about numbers, but maybe we can accept it as being about physical objects. If numbers are essentially patterns, then identity is hypothetical one-to-one identity between sets of objects.
Numbers are not physical, and not ideas - they are objective and non-sensible [Frege]
     Full Idea: Number is neither spatial and physical, like Mill's pile of pebbles, nor yet subjective like ideas, but non-sensible and objective.
     From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §27)
     A reaction: This doesn't require commitment to full-blown universals, nor to a dualist world of mind. The thinking of the brain moves far away from the areas of sensation, and the brain's capacity for truth is its capacity for objectivity.
Numbers are objects, because they can take the definite article, and can't be plurals [Frege]
     Full Idea: Individual numbers are objects, as is indicated by the use of the definite article in expressions like 'the number two', and by the impossibility of speaking of ones, twos, etc. in the plural.
     From: Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §68 n)
     A reaction: Hm. The beginnings of linguistic philosophy, with all its problems. It is well known that 'for the sake of the children' doesn't make an ontological commitment to 'sakes'. The children might 'enter in threes', but the second half is a good point.
Our concepts recognise existing relations, they don't change them [Frege]
     Full Idea: The bringing of an object under a concept is merely the recognition of a relation which previously already obtained, [but in the abstractionist view] objects are essentially changed by the process, so that objects brought under a concept become similar.
     From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.324)
     A reaction: Frege's view would have to account for occasional misapplications of concepts, like taking a dolphin to be a fish, or falsely thinking there is someone in the cellar.
Numbers are not real like the sea, but (crucially) they are still objective [Frege]
     Full Idea: The sea is something real and a number is not; but this does not prevent it from being something objective; and that is the important thing.
     From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.337)
     A reaction: This seems a qualification of Frege's platonism. It is why people start talking about abstract items which 'subsist', instead of 'exist'. It shows Frege's motivation in all this, which is to secure logic and maths from the vagaries of psychology.
Restricted Platonism is just an ideal projection of a domain of thought [Bernays]
     Full Idea: A restricted Platonism does not claim to be more than, so to speak, an ideal projection of a domain of thought.
     From: Paul Bernays (On Platonism in Mathematics [1934], p.261)
     A reaction: I have always found Platonism to be congenial when it talks of 'ideals', and ridiculous when it talks of a special form of 'existence'. Ideals only 'exist' because we idealise things. I may declare myself, after all, to be a Restricted Platonist.
Mathematical objects are as essential as physical objects are for perception [Gödel]
     Full Idea: Classes and concepts may be conceived of as real objects, ..and are as necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions, with neither case being about 'data'.
     From: Kurt Gödel (Russell's Mathematical Logic [1944], p.456)
     A reaction: Note that while he thinks real objects are essential for mathematics, be may not be claiming the same thing for our knowledge of logic. If logic contains no objects, then how could mathematics be reduced to it, as in logicism?
Mathematics isn't surprising, given that we experience many objects as abstract [Boolos]
     Full Idea: It is no surprise that we should be able to reason mathematically about many of the things we experience, for they are already 'abstract'.
     From: George Boolos (Must We Believe in Set Theory? [1997], p.129)
     A reaction: He has just given a list of exemplary abstract objects (Idea 10489), but I think there is a more interesting idea here - that our experience of actual physical objects is to some extent abstract, as soon as it is conceptualised.
Platonists like axioms and decisions, Aristotelians like definitions, possibilities and logic [Badiou]
     Full Idea: A Platonist's interest focuses on axioms in which the decision of thought is played out, where an Aristotelian or Leibnizian interest focuses on definitions laying out the representation of possibilities (...and the essence of mathematics is logic).
     From: Alain Badiou (Briefings on Existence [1998], 7)
     A reaction: See Idea 12323 for the significance of the Platonist approach. So logicism is an Aristotelian project? Frege is not a true platonist? I like the notion of 'the representation of possibilities', so will vote for the Aristotelians, against Badiou.
Number platonism says that natural number is a sortal concept [Wright,C]
     Full Idea: Number-theoretic platonism is just the thesis that natural number is a sortal concept.
     From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.i)
     A reaction: See Crispin Wright on sortals to expound this. An odd way to express platonism, but he is presenting the Fregean version of it.
It is claimed that numbers are objects which essentially represent cardinality quantifiers [Hodes]
     Full Idea: The mathematical object-theorist says a number is an object that represents a cardinality quantifier, with the representation relation as the entire essence of the nature of such objects as cardinal numbers like 4.
     From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984])
     A reaction: [compressed] This a classic case of a theory beginning to look dubious once you spell it our precisely. The obvious thought is to make do with the numerical quantifiers, and dispense with the objects. Do other quantifiers need objects to support them?
Numerical terms can't really stand for quantifiers, because that would make them first-level [Hodes]
     Full Idea: The dogmatic Frege is more right than wrong in denying that numerical terms can stand for numerical quantifiers, for there cannot be a language in which object-quantifiers and objects are simultaneously viewed as level zero.
     From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.142)
     A reaction: Subtle. We see why Frege goes on to say that numbers are level zero (i.e. they are objects). We are free, it seems, to rewrite sentences containing number terms to suit whatever logical form appeals. Numbers are just quantifiers?
The Indispensability Argument is the only serious ground for the existence of mathematical entities [Field,H]
     Full Idea: There is one and only one serious argument for the existence of mathematical entities, and that is the Indispensability Argument of Putnam and Quine.
     From: Hartry Field (Science without Numbers [1980], p.5), quoted by Stewart Shapiro - Thinking About Mathematics 9.1
     A reaction: Personally I don't believe (and nor does Field) that this gives a good enough reason to believe in such things. Quine (who likes 'desert landscapes' in ontology) ends up believing that sets are real because of his argument. Not for me.
We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro]
     Full Idea: We must distinguish between 'realism in ontology' - that mathematical objects exist - and 'realism in truth-value', which is suggested by the model-theoretic framework - that each well-formed meaningful sentence is non-vacuously either true or false.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
     A reaction: My inclination is fairly strongly towards realism of the second kind, but not of the first. A view about the notion of a 'truth-maker' might therefore be required. What do the truths refer to? Answer: not objects, but abstractions from objects.
If mathematical objects are accepted, then a number of standard principles will follow [Shapiro]
     Full Idea: One who believes in the independent existence of mathematical objects is likely to accept the law of excluded middle, impredicative definitions, the axiom of choice, extensionality, and arbitrary sets and functions.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 1)
     A reaction: The underlying thought is that since the objects pre-exist, all of the above simply describe the relations between them, rather than having to actually bring the objects into existence. Personally I would seek a middle ground.
Platonists claim we can state the essence of a number without reference to the others [Shapiro]
     Full Idea: The Platonist view may be that one can state the essence of each number, without referring to the other numbers.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.1)
     A reaction: Frege certainly talks this way (in his 'borehole' analogy). Fine, we are asked to spell out the essence of some number, without making reference either to any 'units' composing it, or to any other number adjacent to it or composing it. Reals?
Platonism must accept that the Peano Axioms could all be false [Shapiro]
     Full Idea: A traditional Platonist has to face the possibility that all of the Peano Axioms are false.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.7)
     A reaction: This would be because the objects exist independently, and so the Axioms are a mere human attempt at pinning them down. For the Formalist the axioms create the numbers, and so couldn't be false. This makes me, alas, warm to platonism!
Sets are instances of numbers (rather than 'collections'); numbers explain sets, not vice versa [Lowe]
     Full Idea: I favour an account of sets which sees them as being instances of numbers, thereby avoiding the unhelpful metaphor which speaks of a set as being a 'collection' of things. This reverses the normal view, which explains numbers in terms of sets.
     From: E.J. Lowe (The Possibility of Metaphysics [1998], 10)
     A reaction: Cf. Idea 8297. Either a set is basic, or a number is. We might graft onto Lowe's view an account of numbers in terms of patterns, which would give an empirical basis to the picture, and give us numbers which could be used to explain sets.
If 2 is a particular, then adding particulars to themselves does nothing, and 2+2=2 [Lowe]
     Full Idea: If 2 is a particular, 'adding' it to itself can, it would seem, only leave us with 2, not another number. (If 'Socrates + Socrates' denotes anything, it most plausibly just denotes Socrates).
     From: E.J. Lowe (The Possibility of Metaphysics [1998], 10.7)
     A reaction: This suggest Kant's claim that arithmetical sums are synthetic (Idea 5558). It is a nice question why, when you put two 2s together, they come up with something new. Addition is movement. Among patterns, or along abstract sequences.
The irrationality of root-2 was achieved by intellect, not experience [Brown,JR]
     Full Idea: We could not discover irrational numbers by physical measurement. The discovery of the irrationality of the square root of two was an intellectual achievement, not at all connected to sense experience.
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 1)
     A reaction: Brown declares himself a platonist, and this is clearly a key argument for him, and rather a good one. Hm. I'll get back to you on this one...
If there are infinite numbers and finite concrete objects, this implies that numbers are abstract objects [Lowe]
     Full Idea: The Peano postulates imply an infinity of numbers, but there are probably not infinitely many concrete objects in existence, so natural numbers must be abstract objects.
     From: E.J. Lowe (A Survey of Metaphysics [2002], p.375)
     A reaction: Presumably they are abstract objects even if they aren't universals. 'Abstract' is an essential term in our ontological vocabulary to cover such cases. Perhaps possible concrete objects are infinite.
Platonism claims that some true assertions have singular terms denoting abstractions, so abstractions exist [Williamson]
     Full Idea: The Fregean argument for platonism is that some true assertions contain singular terms which denote abstract objects if they denote anything; since the assertions are true, the singular terms denote.
     From: Timothy Williamson (Review of Bob Hale's 'Abstract Objects' [1988])
     A reaction: I am perplexed that anyone would rest their view of reality on such an argument. The obvious comparison would be with true remarks about blatantly fictional characters, or blatantly invented concepts such as 'checkmate'.
Why is arithmetic hard to learn, but then becomes easy? [Hofweber]
     Full Idea: Why is arithmetic so hard to learn, and why does it seem so easy to us now? For example, subtracting 789 from 26,789.
     From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §4.2)
     A reaction: His answer that we find thinking about objects very easy, but as children we have to learn with difficulty the conversion of the determiner/adjectival number words, so that we come to think of them as objects.
If 'there are red roses' implies 'there are roses', then 'there are prime numbers' implies 'there are numbers' [Schaffer,J]
     Full Idea: We can automatically infer 'there are roses' from 'there are red roses' (with no shift in the meaning of 'roses'). Likewise one can automatically infer 'there are numbers' from 'there are prime numbers'.
     From: Jonathan Schaffer (On What Grounds What [2009], 2.1)
     A reaction: He similarly observes that the atheist's 'God is a fictional character' implies 'there are fictional characters'. Schaffer is not committing to a strong platonism with his claim - merely that the existence of numbers is hardly worth disputing.
We can only mentally construct potential infinities, but maths needs actual infinities [Hossack]
     Full Idea: Numbers cannot be mental objects constructed by our own minds: there exists at most a potential infinity of mental constructions, whereas the axioms of mathematics require an actual infinity of numbers.
     From: Keith Hossack (Knowledge and the Philosophy of Number [2020], Intro 2)
     A reaction: Doubt this, but don't know enough to refute it. Actual infinities were a fairly late addition to maths, I think. I would think treating fictional complete infinities as real would be sufficient for the job. Like journeys which include imagined roads.