| 2007 | Introducing the Philosophy of Mathematics |
| p.128 | 8713 | In classical/realist logic the connectives are defined by truth-tables | |
| Full Idea: In the classical or realist view of logic the meaning of abstract symbols for logical connectives is given by the truth-tables for the symbol. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007]) | |||
| A reaction: Presumably this is realist because it connects them to 'truth', but only if that involves a fairly 'realist' view of truth. You could, of course, translate 'true' and 'false' in the table to empty (formalist) symbols such a 0 and 1. Logic is electronics. |
| 1.4 | p.12 | 8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction |
| Full Idea: The natural numbers are quite primitive, and are what we first learn about. The order of objects (the 'ordinals') is one level of abstraction up from the natural numbers: we impose an order on objects. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4) | |||
| A reaction: Note the talk of 'levels of abstraction'. So is there a first level of abstraction? Dedekind disagrees with Friend (Idea 7524). I would say that natural numbers are abstracted from something, but I'm not sure what. See Structuralism in maths. |
| 1.4 | p.13 | 8663 | Raising omega to successive powers of omega reveal an infinity of infinities |
| Full Idea: After the multiples of omega, we can successively raise omega to powers of omega, and after that is done an infinite number of times we arrive at a new limit ordinal, which is called 'epsilon'. We have an infinite number of infinite ordinals. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4) | |||
| A reaction: When most people are dumbstruck by the idea of a single infinity, Cantor unleashes an infinity of infinities, which must be the highest into the stratosphere of abstract thought that any human being has ever gone. |
| 1.4 | p.13 | 8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega |
| Full Idea: The first 'limit ordinal' is called 'omega', which is ordinal because it is greater than other numbers, but it has no immediate predecessor. But it has successors, and after all of those we come to twice-omega, which is the next limit ordinal. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4) | |||
| A reaction: This is the gateway to Cantor's paradise of infinities, which Hilbert loved and defended. Who could resist the pleasure of being totally boggled (like Aristotle) by a concept such as infinity, only to have someone draw a map of it? See 8663 for sequel. |
| 1.5 | p.14 | 8664 | Cardinal numbers answer 'how many?', with the order being irrelevant |
| Full Idea: The 'cardinal' numbers answer the question 'How many?'; the order of presentation of the objects being counted as immaterial. Def: the cardinality of a set is the number of members of the set. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |||
| A reaction: If one asks whether cardinals or ordinals are logically prior (see Ideas 7524 and 8661), I am inclined to answer 'neither'. Presenting them as answers to the questions 'how many?' and 'which comes first?' is illuminating. |
| 1.5 | p.15 | 8666 | Infinite sets correspond one-to-one with a subset |
| Full Idea: Two sets are the same size if they can be placed in one-to-one correspondence. But even numbers have one-to-one correspondence with the natural numbers. So a set is infinite if it has one-one correspondence with a proper subset. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |||
| A reaction: Dedekind's definition. We can match 1 with 2, 2 with 4, 3 with 6, 4 with 8, etc. Logicians seem happy to give as a definition anything which fixes the target uniquely, even if it doesn't give the essence. See Frege on 0 and 1, Ideas 8653/4. |
| 1.5 | p.15 | 8665 | A 'proper subset' of A contains only members of A, but not all of them |
| Full Idea: A 'subset' of A is a set containing only members of A, and a 'proper subset' is one that does not contain all the members of A. Note that the empty set is a subset of every set, but it is not a member of every set. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |||
| A reaction: Is it the same empty set in each case? 'No pens' is a subset of 'pens', but is it a subset of 'paper'? Idea 8219 should be borne in mind when discussing such things, though I am not saying I agree with it. |
| 1.5 | p.16 | 8667 | The 'integers' are the positive and negative natural numbers, plus zero |
| Full Idea: The set of 'integers' is all of the negative natural numbers, and zero, together with the positive natural numbers. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |||
| A reaction: Zero always looks like a misfit at this party. Credit and debit explain positive and negative nicely, but what is the difference between having no money, and money being irrelevant? I can be 'broke', but can the North Pole be broke? |
| 1.5 | p.17 | 8668 | The 'rational' numbers are those representable as fractions |
| Full Idea: The 'rational' numbers are all those that can be represented in the form m/n (i.e. as fractions), where m and n are natural numbers different from zero. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |||
| A reaction: Pythagoreans needed numbers to stop there, in order to represent the whole of reality numerically. See irrational numbers for the ensuing disaster. How can a universe with a finite number of particles contain numbers that are not 'rational'? |
| 1.5 | p.17 | 8669 | Between any two rational numbers there is an infinite number of rational numbers |
| Full Idea: Since between any two rational numbers there is an infinite number of rational numbers, we could consider that we have infinity in three dimensions: positive numbers, negative numbers, and the 'depth' of infinite numbers between any rational numbers. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |||
| A reaction: This is before we even reach Cantor's staggering infinities (Ideas 8662 and 8663), which presumably reside at the outer reaches of all three of these dimensions of infinity. The 'deep' infinities come from fractions with huge denominators. |
| 1.5 | p.19 | 8670 | A number is 'irrational' if it cannot be represented as a fraction |
| Full Idea: A number is 'irrational' just in case it cannot be represented as a fraction. An irrational number has an infinite non-repeating decimal expansion. Famous examples are pi and e. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |||
| A reaction: There must be an infinite number of irrational numbers. You could, for example, take the expansion of pi, and change just one digit to produce a new irrational number, and pi has an infinity of digits to tinker with. |
| 1.5 | p.19 | 8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps |
| Full Idea: The set of 'real' numbers, which consists of the rational numbers and the irrational numbers together, represents "the continuum", since it is like a smooth line which has no gaps (unlike the rational numbers, which have the irrationals missing). | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |||
| A reaction: The Continuum is the perfect abstract object, because a series of abstractions has arrived at a vast limit in its nature. It still has dizzying infinities contained within it, and at either end of the line. It makes you feel humble. |
| 1.5 | p.21 | 8672 | A 'powerset' is all the subsets of a set |
| Full Idea: The 'powerset' of a set is a set made up of all the subsets of a set. For example, the powerset of {3,7,9} is {null, {3}, {7}, {9}, {3,7}, {3,9}, {7,9}, {3,7,9}}. Taking the powerset of an infinite set gets us from one infinite cardinality to the next. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |||
| A reaction: Note that the null (empty) set occurs once, but not in the combinations. I begin to have queasy sympathies with the constructivist view of mathematics at this point, since no one has the time, space or energy to 'take' an infinite powerset. |
| 2.3 | p.26 | 8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal |
| Full Idea: The Burali-Forti paradox says that if ordinals are defined by 'gathering' all their predecessors with the empty set, then is the set of all ordinals an ordinal? It is created the same way, so it should be a further member of this 'complete' set! | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |||
| A reaction: This is an example (along with Russell's more famous paradox) of the problems that began to appear in set theory in the early twentieth century. See Idea 8675 for a modern solution. |
| 2.3 | p.27 | 8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' |
| Full Idea: The realist meets the Burali-Forti paradox by saying that all the ordinals are a 'class', not a set. A proper class is what we discuss when we say "all" the so-and-sos when they cannot be reached by normal set-construction. Grammar is their only limit. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |||
| A reaction: This strategy would be useful for Class Nominalism, which tries to define properties in terms of classes, but gets tangled in paradoxes. But why bother with strict sets if easy-going classes will do just as well? Descartes's Dream: everything is rational. |
| 2.3 | p.29 | 8676 | Is mathematics based on sets, types, categories, models or topology? |
| Full Idea: Successful competing founding disciplines in mathematics include: the various set theories, type theory, category theory, model theory and topology. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |||
| A reaction: Or none of the above? Set theories are very popular. Type theory is, apparently, discredited. Shapiro has a version of structuralism based on model theory (which sound promising). Topology is the one that intrigues me... |
| 2.3 | p.32 | 8677 | Set theory makes a minimum ontological claim, that the empty set exists |
| Full Idea: As a realist choice of what is basic in mathematics, set theory is rather clever, because it only makes a very simple ontological claim: that, independent of us, there exists the empty set. The whole hierarchy of finite and infinite sets then follows. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |||
| A reaction: Even so, for non-logicians the existence of the empty set is rather counterintuitive. "There was nobody on the road, so I overtook him". See Ideas 7035 and 8322. You might work back to the empty set, but how do you start from it? |
| 2.3 | p.33 | 8678 | Most mathematical theories can be translated into the language of set theory |
| Full Idea: Most of mathematics can be faithfully redescribed by classical (realist) set theory. More precisely, we can translate other mathematical theories - such as group theory, analysis, calculus, arithmetic, geometry and so on - into the language of set theory. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |||
| A reaction: This is why most mathematicians seem to regard set theory as foundational. We could also translate football matches into the language of atomic physics. |
| 2.3 | p.34 | 3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction |
| Full Idea: Reductio ad absurdum arguments are ones that start by denying what one wants to prove. We then prove a contradiction from this 'denied' idea and more reasonable ideas in one's theory, showing that we were wrong in denying what we wanted to prove. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |||
| A reaction: This is a mathematical definition, which rests on logical contradiction, but in ordinary life (and philosophy) it would be enough to show that denial led to absurdity, rather than actual contradiction. |
| 2.4 | p.36 | 8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects |
| Full Idea: In classical logic definitions are thought of as revealing our attempts to refer to objects, ...but for intuitionist or constructivist logics, if our definitions do not uniquely characterize an object, we are not entitled to discuss the object. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.4) | |||
| A reaction: In defining a chess piece we are obviously creating. In defining a 'tree' we are trying to respond to fact, but the borderlines are vague. Philosophical life would be easier if we were allowed a mixture of creation and fact - so let's have that. |
| 2.5 | p.36 | 8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? |
| Full Idea: The main philosophical problem with the position of platonism or realism is the epistemic problem: of explaining what perception or intuition consists in; how it is possible that we should accurately detect whatever it is we are realists about. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.5) | |||
| A reaction: The best bet, I suppose, is that the mind directly perceives concepts just as eyes perceive the physical (see Idea 8679), but it strikes me as implausible. If we have to come up with a special mental faculty for an area of knowledge, we are in trouble. |
| 2.6 | p.42 | 8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity |
| Full Idea: Zermelo-Fraenkel and Gödel-Bernays set theory differ over the notions of ordinal construction and over the notion of class, among other things. Then there are optional axioms which can be attached, such as the axiom of choice and the axiom of infinity. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.6) | |||
| A reaction: This summarises the reasons why we cannot just talk about 'set theory' as if it was a single concept. The philosophical interest I would take to be found in disentangling the ontological commitments of each version. |
| 3.1 | p.51 | 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them |
| Full Idea: In the hierarchy of reduction, when we investigate questions in biology, we have to assume the laws of chemistry but not of economics. We could never find a law of biology that contradicted something in physics or in chemistry. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 3.1) | |||
| A reaction: This spells out the idea that there is a direction of dependence between aspects of the world, though we should be cautious of talking about 'levels' (see Idea 7003). We cannot choose the direction in which reduction must go. |
| 3.4 | p.64 | 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) |
| Full Idea: The extensional presentation of a concept is just a list of the objects falling under the concept. In contrast, an intensional presentation of a concept gives a characterization of the concept, which allows us to pick out which objects fall under it. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 3.4) | |||
| A reaction: Logicians seem to favour the extensional view, because (in the standard view) sets are defined simply by their members, so concepts can be explained using sets. I take this to be a mistake. The intensional view seems obviously prior. |
| 3.7 | p.77 | 8694 | Free logic was developed for fictional or non-existent objects |
| Full Idea: Free logic is especially designed to help regiment our reasoning about fictional objects, or nonexistent objects of some sort. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 3.7) | |||
| A reaction: This makes it sound marginal, but I wonder whether existential commitment shouldn't be eliminated from all logic. Why do fictional objects need a different logic? What logic should we use for Robin Hood, if we aren't sure whether or not he is real? |
| 4.1 | p.82 | 8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves |
| Full Idea: According to the structuralist, mathematicians study the concepts (objects of study) such as variable, greater, real, add, similar, infinite set, which are one level of abstraction up from prima facie base objects such as numbers, shapes and lines. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.1) | |||
| A reaction: This still seems to imply an ontology in which numbers, shapes and lines exist. I would have thought you could eliminate the 'base objects', and just say that the concepts are one level of abstraction up from the physical world. |
| 4.1 | p.82 | 8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential |
| Full Idea: Structuralism says we study whole structures: objects together with their predicates, relations that bear between them, and functions that take us from one domain of objects to a range of other objects. The objects can even be eliminated. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.1) | |||
| A reaction: The unity of object and predicate is a Quinean idea. The idea that objects are inessential is the dramatic move. To me the proposal has very strong intuitive appeal. 'Eight' is meaningless out of context. Ordinality precedes cardinality? Ideas 7524/8661. |
| 4.4 | p.90 | 8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? |
| Full Idea: Structuralists disagree over whether objects in structures are 'ante rem' (before reality, existing independently of whether the objects exist) or 'in re' (in reality, grounded in the real world, usually in our theories of physics). | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |||
| A reaction: Shapiro holds the first view, Hellman and Resnik the second. The first view sounds too platonist and ontologically extravagant; the second sounds too contingent and limited. The correct account is somewhere in abstractions from the real. |
| 4.4 | p.91 | 8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting |
| Full Idea: In the 'in re' version of mathematical structuralism, pattern-spotting is the process of abstraction. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |||
| A reaction: This might work for non-mathematical abstraction as well, if we are allowed to spot patterns within sensual experience, and patterns within abstractions. Properties are causal patterns in the world? No - properties cause patterns. |
| 4.4 | p.93 | 8701 | The number 8 in isolation from the other numbers is of no interest |
| Full Idea: There is no interest for the mathematician in studying the number 8 in isolation from the other numbers. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |||
| A reaction: This is a crucial and simple point (arising during a discussion of Shapiro's structuralism). Most things are interesting in themselves, as well as for their relationships, but mathematical 'objects' just are relationships. |
| 4.4 | p.93 | 8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent |
| Full Idea: Structuralists give a historical account of why the 'same' number occupies different structures. Numbers are equivalent rather than identical. 8 is the immediate predecessor of 9 in the whole numbers, but in the rationals 9 has no predecessor. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |||
| A reaction: I don't become a different person if I move from a detached house to a terraced house. This suggests that 8 can't be entirely defined by its relations, and yet it is hard to see what its intrinsic nature could be, apart from the units which compose it. |
| 4.5 | p.97 | 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' |
| Full Idea: What the mathematician labels an 'object' in her discipline, is called 'a place in a structure' by the structuralist. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.5) | |||
| A reaction: This is a strategy for dispersing the idea of an object in the world of thought, parallel to attempts to eliminate them from physical ontology (e.g. Idea 614). |
| 5.1 | p.104 | 8705 | Anti-realists see truth as our servant, and epistemically contrained |
| Full Idea: For the anti-realist, truth belongs to us, it is our servant, and as such, it must be 'epistemically constrained'. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.1) | |||
| A reaction: Put as clearly as this, it strikes me as being utterly and spectacularly wrong, a complete failure to grasp the elementary meaning of a concept etc. etc. If we aren't the servants of truth then we jolly we ought to be. Truth is above us. |
| 5.1 | p.106 | 8706 | Constructivism rejects too much mathematics |
| Full Idea: Too much of mathematics is rejected by the constructivist. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.1) | |||
| A reaction: This was Hilbert's view. This seems to be generally true of verificationism. My favourite example is that legitimate speculations can be labelled as meaningless. |
| 5.2 | p.106 | 8707 | Intuitionists typically retain bivalence but reject the law of excluded middle |
| Full Idea: An intuitionist typically retains bivalence, but rejects the law of excluded middle. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2) | |||
| A reaction: The idea would be to say that only T and F are available as truth-values, but failing to be T does not ensure being F, but merely not-T. 'Unproven' is not-T, but may not be F. |
| 5.2 | p.107 | 8708 | Double negation elimination is not valid in intuitionist logic |
| Full Idea: In intuitionist logic, if we do not know that we do not know A, it does not follow that we know A, so the inference (and, in general, double negation elimination) is not intuitionistically valid. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2) | |||
| A reaction: That inference had better not be valid in any logic! I am unaware of not knowing the birthday of someone I have never heard of. Propositional attitudes such as 'know' are notoriously difficult to explain in formal logic. |
| 5.2 | p.108 | 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false |
| Full Idea: The law of excluded middle is purely syntactic: it says for any well-formed formula A, either A or not-A. It is not a semantic law; it does not say that either A is true or A is false. The semantic version (true or false) is the law of bivalence. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2) | |||
| A reaction: No wonder these two are confusing, sufficiently so for a lot of professional philosophers to blur the distinction. Presumably the 'or' is exclusive. So A-and-not-A is a contradiction; but how do you explain a contradiction without mentioning truth? |
| 5.5 | p.122 | 8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." |
| Full Idea: In the intuitionist version of quantification, the universal quantifier (normally read as "all") is understood as "we have a procedure for checking every" or "we have checked every". | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.5) | |||
| A reaction: It seems better to describe this as 'verificationist' (or, as Dummett prefers, 'justificationist'). Intuition suggests an ability to 'see' beyond the evidence. It strikes me as bizarre to say that you can't discuss things you can't check. |
| 6.1 | p.128 | 8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory |
| Full Idea: Central to naturalism about mathematics are 'indispensability arguments', to the effect that some part of mathematics is indispensable to our best physical theory, and therefore we ought to take that part of mathematics to be true. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 6.1) | |||
| A reaction: Quine and Putnam hold this view; Field challenges it. It has the odd consequence that the dispensable parts (if they can be identified!) do not need to be treated as true (even though they might follow logically from the dispensable parts!). Wrong! |
| 6.6 | p.149 | 8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research |
| Full Idea: There are not enough constraints in the Formalist view of mathematics, so there is no way to select a direction for trying to develop mathematics. There is no part of mathematics that is more important than another. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 6.6) | |||
| A reaction: One might reply that an area of maths could be 'important' if lots of other areas depended on it, and big developments would ripple big changes through the interior of the subject. Formalism does, though, seem to reduce maths to a game. |
| Glossary | p.172 | 8721 | An 'impredicative' definition seems circular, because it uses the term being defined |
| Full Idea: An 'impredicative' definition is one that uses the terms being defined in order to give the definition; in some way the definition is then circular. | |||
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], Glossary) | |||
| A reaction: There has been a big controversy in the philosophy of mathematics over these. Shapiro gives the definition of 'village idiot' (which probably mentions 'village') as an example. |