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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers

[general ideas concerning numbers]

26 ideas
We perceive number by the denial of continuity [Aristotle]
     Full Idea: Number we perceive by the denial of continuity.
     From: Aristotle (De Anima [c.329 BCE], 425a20)
     A reaction: This is a key thought. A being which (call it 'Parmenides') which saying all Being as One, would make no distinctions of identity, and so could not count anything. Why would they want numbers?
Pluralities divide into discontinous countables; magnitudes divide into continuous things [Aristotle]
     Full Idea: A plurality is a denumerable quantity, and a magnitude is a measurable quantity. A plurality is what is potentially divisible into things that are not continuous, whereas what is said to be a magnitude is divisible into continuous things.
     From: Aristotle (Metaphysics [c.324 BCE], 1020a09)
     A reaction: This illuminating distinction is basic to the Greek attitude to number, and echoes the distinction between natural and real numbers.
Perhaps numbers are substances? [Aristotle]
     Full Idea: We should consider whether there is some other sort of substance, such as, perhaps, numbers.
     From: Aristotle (Metaphysics [c.324 BCE], 1037a11)
     A reaction: I don't think Aristotle considers numbers to be substances, but Pythagoreans seem to think that way, if they think the world is literally made of numbers.
We can talk of 'innumerable number', about the infinite points on a line [Newton]
     Full Idea: If any man shall take the words number and sum in a larger sense, to understand things which are numberless and sumless (such as the infinite points on a line), I could allow him the contradictious phrase 'innumerable number' without absurdity.
     From: Isaac Newton (Letters to Bentley [1692], 1693.02.25)
     A reaction: [compressed] I take the key point here to be the phrase of taking number 'in a larger sense'. Like the word 'atom' in physics, the word 'number' retains its traditional reference, but has considerably shifted its scope. Amateurs must live with this.
Numbers are formed by addition of units in time [Kant]
     Full Idea: Arithmetic forms its own concepts of numbers by successive addition of units in time.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284)
     A reaction: It is hard to imagine any modern philosopher of mathematics embracing this idea. It sounds as if Kant thinks counting is the foundation of arithmetic, which I quite like.
Numbers are free creations of the human mind, to understand differences [Dedekind]
     Full Idea: Numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things.
     From: Richard Dedekind (Nature and Meaning of Numbers [1888], Pref)
     A reaction: Does this fit real numbers and complex numbers, as well as natural numbers? Frege was concerned by the lack of objectivity in this sort of view. What sort of arithmetic might the Martians have created? Numbers register sameness too.
Numbers enable us to manage the world - to the limits of counting [Nietzsche]
     Full Idea: Numbers are our major means of making the world manageable. We comprehend as far as we can count, i.e. as far as a constancy can be perceived.
     From: Friedrich Nietzsche (Fragments from 1885-1886 [1886], 34[058])
     A reaction: I don't agree with 'major', but it is a nice thought. The intermediate concept is a 'unit', which means identifying something as a 'thing', which is how we seem to grasp the world. So to what extent do we comprehend the infinite. Enter Cantor…
Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order [Benacerraf]
     Full Idea: Not all numbers could possibly have been learned ŕ la Frege-Russell, because we could not have performed that many distinct acts of abstraction. Somewhere along the line a rule had to come in to enable us to obtain more numbers, in the natural order.
     From: Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.165)
     A reaction: Follows on from Idea 13411. I'm not sure how Russell would deal with this, though I am sure his account cannot be swept aside this easily. Nevertheless this seems powerful and convincing, approaching the problem through the epistemology.
We must explain how we know so many numbers, and recognise ones we haven't met before [Benacerraf]
     Full Idea: Both ordinalists and cardinalists, to account for our number words, have to account for the fact that we know so many of them, and that we can 'recognize' numbers which we've neither seen nor heard.
     From: Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.166)
     A reaction: This seems an important contraint on any attempt to explain numbers. Benacerraf is an incipient structuralist, and here presses the importance of rules in our grasp of number. Faced with 42,578,645, we perform an act of deconstruction to grasp it.
Numbers can't be sets if there is no agreement on which sets they are [Benacerraf]
     Full Idea: The fact that Zermelo and Von Neumann disagree on which particular sets the numbers are is fatal to the view that each number is some particular set.
     From: Paul Benacerraf (What Numbers Could Not Be [1965], II)
     A reaction: I agree. A brilliantly simple argument. There is the possibility that one of the two accounts is correct (I would vote for Zermelo), but it is not actually possible to prove it.
There are no such things as numbers [Benacerraf]
     Full Idea: There are no such things as numbers.
     From: Paul Benacerraf (What Numbers Could Not Be [1965], IIIC)
     A reaction: Mill said precisely the same (Idea 9794). I think I agree. There has been a classic error of reification. An abstract pattern is not an object. If I coin a word for all the three-digit numbers in our system, I haven't created a new 'object'.
There is no single unified definition of number [Badiou]
     Full Idea: Apparently - and this is quite unlike old Greek times - there is no single unified definition of number.
     From: Alain Badiou (Briefings on Existence [1998], 11)
Numbers are for measuring and for calculating (and the two must be consistent) [Badiou]
     Full Idea: Number is an instance of measuring (distinguishing the more from the less, and calibrating data), ..and a figure for calculating (one counts with numbers), ..and it ought to be a figure of consistency (the compatibility of order and calculation).
     From: Alain Badiou (Briefings on Existence [1998], 11)
For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P]
     Full Idea: For prime numbers we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))). That is, the only way to multiply two numbers and a get a prime is if one of them is 1.
     From: Peter Smith (Intro to Gödel's Theorems [2007], 04.5)
Number theory aims at the essence of natural numbers, giving their nature, and the epistemology [Wright,C]
     Full Idea: In the Fregean view number theory is a science, aimed at those truths furnished by the essential properties of zero and its successors. The two broad question are then the nature of the objects, and the epistemology of those facts.
     From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro)
     A reaction: [compressed] I pounce on the word 'essence' here (my thing). My first question is about the extent to which the natural numbers all have one generic essence, and the extent to which they are individuals (bless their little cotton socks).
Mathematics is higher-order modal logic [Hodes]
     Full Idea: I take the view that (agreeing with Aristotle) mathematics only requires the notion of a potential infinity, ...and that mathematics is higher-order modal logic.
     From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984])
     A reaction: Modern 'modal' accounts of mathematics I take to be heirs of 'if-thenism', which seems to have been Russell's development of Frege's original logicism. I'm beginning to think it is right. But what is the subject-matter of arithmetic?
In Field's version of science, space-time points replace real numbers [Field,H, by Szabó]
     Full Idea: Field's nominalist version of science develops a version of Newtonian gravitational theory, where no quantifiers range over mathematical entities, and space-time points and regions play the role of surrogates for real numbers.
     From: report of Hartry Field (Science without Numbers [1980]) by Zoltán Gendler Szabó - Nominalism 5.1
     A reaction: This seems to be a very artificial contrivance, but Field has launched a programme for rewriting science so that numbers can be omitted. All of this is Field's rebellion against the Indispensability Argument for mathematics. I sympathise.
If 'the number of Democrats is on the rise', does that mean that 50 million is on the rise? [Yablo]
     Full Idea: If someone says 'the number of Democrats is on the rise', he or she wants to focus on Democrats, not numbers. If the number is 50 million, is 50 million really on the rise?
     From: Stephen Yablo (Apriority and Existence [2000], §14)
     A reaction: This is a very nice warning from Yablo, against easy platonism, or any sort of platonism at all. We routinely say that numbers are 'increasing', but the real meaning needs entangling. Here it refers to people joining a party.
We should talk about possible existence, rather than actual existence, of numbers [Burgess/Rosen]
     Full Idea: The modal strategy for numbers is to replace assumptions about the actual existence of numbers by assumptions about the possible existence of numbers
     From: JP Burgess / G Rosen (A Subject with No Object [1997], II.B.3.a)
     A reaction: This seems to be quite a good way of dealing with very large numbers and infinities. It is not clear whether 5 is so regularly actualised that we must consider it as permanent, or whether it is just a prominent permanent possibility.
'There are two apples' can be expressed logically, with no mention of numbers [Brown,JR]
     Full Idea: 'There are two apples' can be recast as 'x is an apple and y is an apple, and x isn't y, and if z is an apple it is the same as x or y', which makes no appeal at all to mathematics.
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 4)
     A reaction: He cites this as the basis of Hartry Field's claim that science can be done without numbers. The logic is ∃x∃y∀z(Ax&Ay&(x¬=y)&(Az→z=x∨z=y)).
The meaning of a number isn't just the numerals leading up to it [Heck]
     Full Idea: My knowing what the number '33' denotes cannot consist in my knowing that it denotes the number of decimal numbers between '1' and '33', because I would know that even if it were in hexadecimal (which I don't know well).
     From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 5)
     A reaction: Obviously you wouldn't understand '33' if you didn't understand what '33 things' meant.
The Aristotelian view is that numbers depend on (and are abstracted from) other things [Oderberg]
     Full Idea: The Aristotelian account of numbers is that their existence depends on the existence of things that are not numbers, ..since numbers are abstractions from the existence of things.
     From: David S. Oderberg (Real Essentialism [2007], 1.2)
     A reaction: This is the deeply unfashionable view to which I am attached. The problem is the status of transfinite, complex etc numbers. They look like fictions to me.
What is the relation of number words as singular-terms, adjectives/determiners, and symbols? [Hofweber]
     Full Idea: There are three different uses of the number words: the singular-term use (as in 'the number of moons of Jupiter is four'), the adjectival (or determiner) use (as in 'Jupiter has four moons'), and the symbolic use (as in '4'). How are they related?
     From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §1)
     A reaction: A classic philosophy of language approach to the problem - try to give the truth-conditions for all three types. The main problem is that the first one implies that numbers are objects, whereas the others do not. Why did Frege give priority to the first?
'2 + 2 = 4' can be read as either singular or plural [Hofweber]
     Full Idea: There are two ways to read to read '2 + 2 = 4', as singular ('two and two is four'), and as plural ('two and two are four').
     From: Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §4.1)
     A reaction: Hofweber doesn't notice that this phenomenon occurs elsewhere in English. 'The team is playing well', or 'the team are splitting up'; it simply depends whether you are holding the group in though as an entity, or as individuals. Important for numbers.
Numbers are used as singular terms, as adjectives, and as symbols [Hofweber]
     Full Idea: Number words have a singular term use, and adjectival (or determiner) use, and the symbolic use. The main question is how they relate to each other.
     From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 05.1)
     A reaction: Thus 'the number four is even', 'there are four moons', and '4 comes after 3'.
The Amazonian Piraha language is said to have no number words [Hofweber]
     Full Idea: The now famous Piraha language, of the Amazon region in Brazil, allegedly has no number words.
     From: Thomas Hofweber (Ontology and the Ambitions of Metaphysics [2016], 05.6)
     A reaction: Two groups can be shown to be of equal cardinality, by one-to-one matching rather than by counting. They could get by using 'equals' (and maybe unequally bigger and unequally smaller), and intuitive feelings for sizes of groups.