display all the ideas for this combination of philosophers
37 ideas
| 10059 | In mathematic we are ignorant of both subject-matter and truth [Russell] |
| Full Idea: Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. | |
| From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.76) | |
| A reaction: A famous remark, though Musgrave is rather disparaging about Russell's underlying reasoning here. |
| 14152 | In geometry, Kant and idealists aimed at the certainty of the premisses [Russell] |
| Full Idea: The approach to practical geometry of the idealists, and especially of Kant, was that we must be certain of the premisses on their own account. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §353) |
| 14154 | Geometry throws no light on the nature of actual space [Russell] |
| Full Idea: Geometry no longer throws any direct light on the nature of actual space. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §353) | |
| A reaction: This was 1903. Minkowski then contributed a geometry of space which was used in Einstein's General Theory. It looks to me as if geometry reveals the possibilities for actual space. |
| 14151 | Pure geometry is deductive, and neutral over what exists [Russell] |
| Full Idea: As a branch of pure mathematics, geometry is strictly deductive, indifferent to the choice of its premises, and to the question of whether there strictly exist such entities. It just deals with series of more than one dimension. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §352) | |
| A reaction: This seems to be the culmination of the seventeenth century reduction of geometry to algebra. Russell admits that there is also the 'study of actual space'. |
| 14153 | In geometry, empiricists aimed at premisses consistent with experience [Russell] |
| Full Idea: The approach to practical geometry of the empiricists, notably Mill, was to show that no other set of premisses would give results consistent with experience. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §353) | |
| A reaction: The modern phrase might be that geometry just needs to be 'empirically adequate'. The empiricists are faced with the possibility of more than one successful set of premisses, and the idealist don't know how to demonstrate truth. |
| 14155 | Two points have a line joining them (descriptive), a distance (metrical), and a whole line (projective) [Russell, by PG] |
| Full Idea: Two points will define the line that joins them ('descriptive' geometry), the distance between them ('metrical' geometry), and the whole of the extended line ('projective' geometry). | |
| From: report of Bertrand Russell (The Principles of Mathematics [1903], §362) by PG - Db (ideas) | |
| A reaction: [a summary of Russell's §362] Projective Geometry clearly has the highest generality, and the modern view seems to make it the master subject of geometry. |
| 14442 | If straight lines were like ratios they might intersect at a 'gap', and have no point in common [Russell] |
| Full Idea: We wish to say that when two straight lines cross each other they have a point in common, but if the series of points on a line were similar to the series of ratios, the two lines might cross in a 'gap' and have no point in common. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], X) | |
| A reaction: You can make a Dedekind Cut in the line of ratios (the rationals), so there must be gaps. I love this idea. We take for granted intersection at a point, but physical lines may not coincide. That abstract lines might fail also is lovely! |
| 18254 | Russell's approach had to treat real 5/8 as different from rational 5/8 [Russell, by Dummett] |
| Full Idea: Russell defined the rationals as ratios of integers, and was therefore forced to treat the real number 5/8 as an object distinct from the rational 5/8. | |
| From: report of Bertrand Russell (The Principles of Mathematics [1903]) by Michael Dummett - Frege philosophy of mathematics 21 'Frege's' |
| 14144 | Ordinals result from likeness among relations, as cardinals from similarity among classes [Russell] |
| Full Idea: Ordinal numbers result from likeness among relations, as cardinals from similarity among classes. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §293) |
| 14438 | New numbers solve problems: negatives for subtraction, fractions for division, complex for equations [Russell] |
| Full Idea: Every generalisation of number has presented itself as needed for some simple problem. Negative numbers are needed to make subtraction always possible; fractions to make division always possible; complex numbers to make solutions of equations possible. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VII) | |
| A reaction: Doesn't this rather suggest that we made them up? If new problems turn up, we'll invent another lot. We already have added 'surreal' numbers. |
| 14128 | Some claim priority for the ordinals over cardinals, but there is no logical priority between them [Russell] |
| Full Idea: It is claimed that ordinals are prior to cardinals, because they form the progression which is relevant to mathematics, but they both form progressions and have the same ordinal properties. There is nothing to choose in logical priority between them. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §230) | |
| A reaction: We have an intuitive notion of the size of a set without number, but you can't actually start counting without number, so the ordering seems to be the key to the business, which (I would have thought) points to ordinals as prior. |
| 14129 | Ordinals presuppose two relations, where cardinals only presuppose one [Russell] |
| Full Idea: Ordinals presuppose serial and one-one relations, whereas cardinals only presuppose one-one relations. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §232) | |
| A reaction: This seems to award the palm to the cardinals, for their greater logical simplicity, but I have already given the award to the ordinals in the previous idea, and I am not going back on that. |
| 14132 | Properties of numbers don't rely on progressions, so cardinals may be more basic [Russell] |
| Full Idea: The properties of number must be capable of proof without appeal to the general properties of progressions, since cardinals can be independently defined, and must be seen in a progression before theories of progression are applied to them. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §243) | |
| A reaction: Russell says there is no logical priority between ordinals and cardinals, but it is simpler to start an account with cardinals. |
| 13510 | Could a number just be something which occurs in a progression? [Russell, by Hart,WD] |
| Full Idea: Russell toyed with the idea that there is nothing to being a natural number beyond occurring in a progression | |
| From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919], p.8) by William D. Hart - The Evolution of Logic 5 | |
| A reaction: How could you define a progression, without a prior access to numbers? - Arrange all the objects in the universe in ascending order of mass. Use scales to make the selection. Hence a finite progression, with no numbers! |
| 14141 | Ordinals are defined through mathematical induction [Russell] |
| Full Idea: The ordinal numbers are defined by some relation to mathematical induction. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §290) |
| 14142 | Ordinals are types of series of terms in a row, rather than the 'nth' instance [Russell] |
| Full Idea: The finite ordinals may be conceived as types of series; ..the ordinal number may be taken as 'n terms in a row'; this is distinct from the 'nth', and logically prior to it. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §290) | |
| A reaction: Worth nothing, because the popular and traditional use of 'ordinal' (as in learning a foreign language) is to mean the nth instance of something, rather than a whole series. |
| 14139 | Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic [Russell] |
| Full Idea: Unlike the transfinite cardinals, the transfinite ordinals do not obey the commutative law, and their arithmetic is therefore quite different from elementary arithmetic. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §290) |
| 14145 | For Cantor ordinals are types of order, not numbers [Russell] |
| Full Idea: In his most recent article Cantor speaks of ordinals as types of order, not as numbers. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §298) | |
| A reaction: Russell likes this because it supports his own view of ordinals as classes of serial relations. It has become orthodoxy to refer to heaps of things as 'numbers' when the people who introduced them may not have seen them that way. |
| 14146 | We aren't sure if one cardinal number is always bigger than another [Russell] |
| Full Idea: We do not know that of any two different cardinal numbers one must be the greater. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §300) | |
| A reaction: This was 1903, and I don't know whether the situation has changed. I find this thought extremely mind-boggling, given that cardinals are supposed to answer the question 'how many?' Presumably they can't be identical either. See Burali-Forti. |
| 14135 | Real numbers are a class of rational numbers (and so not really numbers at all) [Russell] |
| Full Idea: Real numbers are not really numbers at all, but something quite different; ...a real number, so I shall contend, is nothing but a certain class of rational numbers. ...A segment of rationals is a real number. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §258) |
| 14436 | A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum [Russell] |
| Full Idea: There is no maximum to the ratios whose square is less than 2, and no minimum to those whose square is greater than 2. This division of a series into two classes is called a 'Dedekind Cut'. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VII) |
| 14439 | A complex number is simply an ordered couple of real numbers [Russell] |
| Full Idea: A complex number may be regarded and defined as simply an ordered couple of real numbers | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VII) |
| 14421 | Discovering that 1 is a number was difficult [Russell] |
| Full Idea: The discovery that 1 is a number must have been difficult. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I) | |
| A reaction: Interesting that he calls it a 'discovery'. I am tempted to call it a 'decision'. |
| 14123 | Some quantities can't be measured, and some non-quantities are measurable [Russell] |
| Full Idea: Some quantities cannot be measured (such as pain), and some things which are not quantities can be measured (such as certain series). | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §150) |
| 14158 | Quantity is not part of mathematics, where it is replaced by order [Russell] |
| Full Idea: Quantity, though philosophers seem to think it essential to mathematics, does not occur in pure mathematics, and does occur in many cases not amenable to mathematical treatment. The place of quantity is taken by order. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §405) | |
| A reaction: He gives pain as an example of a quantity which cannot be treated mathematically. |
| 14120 | Counting explains none of the real problems about the foundations of arithmetic [Russell] |
| Full Idea: The process of counting gives us no indication as to what the numbers are, as to why they form a series, or as to how it is to be proved that there are n numbers from 1 to n. Hence counting is irrelevant to the foundations of arithmetic. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §129) | |
| A reaction: I take it to be the first truth in the philosophy of mathematics that if there is a system of numbers which won't do the job of counting, then that system is irrelevant. Counting always comes first. |
| 14424 | Numbers are needed for counting, so they need a meaning, and not just formal properties [Russell] |
| Full Idea: We want our numbers to be such as can be used for counting common objects, and this requires that our numbers should have a definite meaning, not merely that they should have certain formal properties. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I) | |
| A reaction: Why would just having certain formal properties be insufficient for counting? You just need an ordered series of unique items. It isn't just that we 'want' this. If you define something that we can't count with, you haven't defined numbers. |
| 14118 | We can define one-to-one without mentioning unity [Russell] |
| Full Idea: It is possible, without the notion of unity, to define what is meant by one-to-one. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §109) | |
| A reaction: This is the trick which enables the Greek account of numbers, based on units, to be abandoned. But when you have arranged the boys and the girls one-to-one, you have not yet got a concept of number. |
| 14441 | The formal laws of arithmetic are the Commutative, the Associative and the Distributive [Russell] |
| Full Idea: The usual formal laws of arithmetic are the Commutative Law [a+b=b+a and axb=bxa], the Associative Law [(a+b)+c=a+(b+c) and (axb)xc=ax(bxc)], and the Distributive Law [a(b+c)=ab+ac)]. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], IX) |
| 14119 | We do not currently know whether, of two infinite numbers, one must be greater than the other [Russell] |
| Full Idea: It is not at present known whether, of two different infinite numbers, one must be greater and the other less. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §118) | |
| A reaction: This must refer to cardinal numbers, as ordinal numbers have an order. The point is that the proper subset is equal to the set (according to Dedekind). |
| 14133 | There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal) [Russell] |
| Full Idea: The theory of infinity has two forms, cardinal and ordinal, of which the former springs from the logical theory of numbers; the theory of continuity is purely ordinal. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §249) |
| 14420 | Infinity and continuity used to be philosophy, but are now mathematics [Russell] |
| Full Idea: The nature of infinity and continuity belonged in former days to philosophy, but belongs now to mathematics. | |
| From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], Pref) | |
| A reaction: It is hard to disagree, since mathematicians since Cantor have revealed so much about infinite numbers (through set theory), but I think it remains an open question whether philosophers have anything distinctive to contribute. |
| 7556 | A collection is infinite if you can remove some terms without diminishing its number [Russell] |
| Full Idea: A collection of terms is infinite if it contains as parts other collections which have as many terms as it has; that is, you can take away some terms of the collection without diminishing its number; there are as many even numbers as numbers all together. | |
| From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.86) | |
| A reaction: He cites Dedekind and Cantor as source for these ideas. If it won't obey the rule that subtraction makes it smaller, then it clearly isn't a number, and really it should be banned from all mathematics. |
| 14134 | Infinite numbers are distinguished by disobeying induction, and the part equalling the whole [Russell] |
| Full Idea: There are two differences of infinite numbers from finite: that they do not obey mathematical induction (both cardinals and ordinals), and that the whole contains a part consisting of the same number of terms (applying only to ordinals). | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §250) |
| 14143 | ω names the whole series, or the generating relation of the series of ordinal numbers [Russell] |
| Full Idea: The ordinal representing the whole series must be different from what represents a segment of itself, with no immediate predecessor, since the series has no last term. ω names the class progression, or generating relation of series of this class. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §291) | |
| A reaction: He is paraphrasing Cantor's original account of ω. |
| 14138 | You can't get a new transfinite cardinal from an old one just by adding finite numbers to it [Russell] |
| Full Idea: It must not be supposed that we can obtain a new transfinite cardinal by merely adding one to it, or even by adding any finite number, or aleph-0. On the contrary, such puny weapons cannot disturb the transfinite cardinals. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §288) | |
| A reaction: If you add one, the original cardinal would be a subset of the new one, and infinite numbers have their subsets equal to the whole, so you have gone nowhere. You begin to wonder whether transfinite cardinals are numbers at all. |
| 14140 | For every transfinite cardinal there is an infinite collection of transfinite ordinals [Russell] |
| Full Idea: For every transfinite cardinal there is an infinite collection of transfinite ordinals, although the cardinal number of all ordinals is the same as or less than that of all cardinals. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §290) | |
| A reaction: Sort that one out, and you are beginning to get to grips with the world of the transfinite! Sounds like there are more ordinals than cardinals, and more cardinals than ordinals. |