Logical Form, Mathematical Practice, and Frege s Begriffsschrift. Danielle Macbeth Haverford College

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1 Logical Form, Mathematical Practice, and Frege s Begriffsschrift Danielle Macbeth Haverford College Logic Colloquium 2015, Helsinki, August 3 8, 2015

2 Outline written mathematics report of reasoning display of reasoning read mechanically (standard reading of Frege s notation) read mathematically Frege s proof of Theorem 133 in the 1879 logic

3 Reporting vs. displaying reasoning You have been given the task of dividing eight hundred and seventy-three by seventeen. But Hindu-Arabic numeration has not yet been invented. And you have no other means, such as a counting board, of working out the solution to the problem. You have to do the problem in your head, by engaging in a stepwise course of mental arithmetic. You perhaps reason as follows.

4 A report of reasoning A hundred seventeens is seventeen hundred, so fifty seventeens is half that: eight hundred and fifty. If we add one more seventeen then we will have fiftyone seventeens and a total of eight hundred and sixtyseven. Eight hundred and sixty-seven is six less than eight hundred and seventy-three, and so the answer is fifty-one with six remainder.

5 A display of reasoning The written positional system of Hindu-Arabic numerals having been devised together with its rules for calculating, we can also solve the problem using paperand-pencil reasoning

6 Reporting vs. displaying reasoning The written English in the report provides a description of some reasoning, a description of the steps a person might go through in order to find, by mental arithmetic, the solution to the problem. The description does not give the steps themselves: to say or write, for example, add one more seventeen is not to add one more seventeen. It is an instruction, not a step of mathematical reasoning. The paper-and-pencil calculation shows the steps in a chain of arithmetical reasoning; it shows how the reasoning goes.

7 That the sum of arbitrarily many even numbers is even: a report The demonstration as reported by Euclid (IX.21): For let as many numbers as we please, AB, BC, CD, DE, be added together; I say the whole AE is even. For since each of the numbers AB, BC, CD, DE, is even, it has a half part (by definition of even); So that the whole also has a half part. But an even number is that which is divisible into two equal parts; Therefore AE is even. QED

8 That the sum of arbitrarily many even numbers is even: a report Euclid simply asserts that since each of the added numbers is even (by hypothesis), and so has a half part, the whole also must have a half part (and therefore is even). The inference from the fact that all the summands have half parts to the fact that the sum has a half part is not made in accordance with any previously stated rule. One simply has to get that it is a valid inference, get that it is a good step of reasoning. The inference is reported rather than displayed.

9 That the sum of arbitrarily many even numbers is even: a display One can display what it is to be an even number in the symbolic language of arithmetic and algebra thus: 2a. One can also display in that system the starting point of the problem, a sum of arbitrarily many even numbers: 2a + 2b + 2c n. Now we apply an antecedently stated rule of the system to get this: 2(a + b + c n). This is clearly an even number, and so we are done.

10 Reporting vs. displaying reasoning Euclid has no means of displaying in a mathematically tractable way what it is to be an even number. Because he does not, Euclid can only report the reasoning involved in proving our little theorem, and either one discerns the goodness of the inference or one does not. But we can display the form of an even number in a complex sign in the symbolic language of arithmetic and algebra. Because we can, we can show the reasoning that is needed to establish the desired result.

11 The Idea of Displaying Reasoning In specially devised systems of written signs one can display what it is to be this or that mathematical entity. And one can do so in a way that is mathematically tractable. That is, one can display what it is to be this or that mathematical entity in a way that enables reasoning in the system of signs according to antecedently specified rules.

12 The distinction of mathematical and mechanical reasoning The claim: Given a system of signs within which to reason, there can be two very different ways to read it, either mathematically or mechanically.

13 A mathematical proof in a Euclidean diagram In Proposition I.1 Euclid shows that an equilateral triangle can be constructed on a given straight line. One begins by drawing a diagram according to the given specifications:

14 A mathematical proof in a Euclidean diagram Now we reason through this diagram to show the result that is wanted, that an equilateral triangle can be constructed on a given straight line.

15 A mathematical proof in a Euclidean diagram Now we reason through this diagram to show the result that is wanted, that an equilateral triangle can be constructed on a given straight line.

16 A mathematical proof in a Euclidean diagram Now we reason through this diagram to show the result that is wanted, that an equilateral triangle can be constructed on a given straight line.

17 A mathematical proof in a Euclidean diagram Now we reason through this diagram to show the result that is wanted, that an equilateral triangle can be constructed on a given straight line.

18 Mechanical vs. mathematical proof As presented in Euclid s Elements, the proof is of course mathematical. The result is necessary, a priori, and completely general. One proves that an equilateral triangle, though not any one in particular, can be constructed on a given straight line, that is, any (given) straight line. A mechanical proof of the (better: a corresponding) result is very different, even though the drawn diagram is, or may be, visually indistinguishable from the one used in the mathematical proof.

19 A mechanical proof Suppose you wanted to construct an actual, particular equilateral triangle of a certain size, orientation, and location. Essentially the same diagram could be used but some of the constraints would be different. If you want an instance then you are best off using straight-edge and compass because, if it is an instance that is wanted, the lines should be as straight as possible, and the circles as circular as possible. Only so will you get something that actually looks, as far as possible, like an equilateral triangle.

20 Mechanical vs. mathematical proof A mechanical proof is aimed at producing an actual instance of something, say, an equilateral triangle. It is best managed with a straight-edge and compass to ensure, as far as possible, that sides are straight and, where needed, equal in length. A mathematical proof does not require such (mechanical) devices as straight-edge and compass. Why not?

21 Mathematical proof A drawn circle in Euclid is not an instance but instead a complex sign formulating what it is to be a circle, namely, a plane figure all points on the circumference of which are equidistant to a center. This sign does not need to look especially circular because it is drawn with the intention of licensing inferences according to the rule that if you have two radii of one circle then you can infer that they are equal in length. Similarly, with the resultant equilateral triangle: it does not need to look much like an equilateral triangle. The point is to show mathematically, by reasoning, that it is one.

22 Mathematical and mechanical proof A Euclidean circle is a plane figure all points on the circumference of which are equidistant from a center. Mathematically, this characterization serves to license inferences: to describe something as a circle is to issue an inference license regarding any two of its radii. Mechanically, the characterization gives necessary and sufficient conditions for something s being a circle. To be a circle mechanically considered is to meet certain specifications, specifications a drawn circle can only approximate.

23 Mathematical reasoning: the basic idea In a Euclidean diagram, the primitive signs lines, points, and so on do not merely stand in for or picture something. Instead they express something like Fregean senses, and together in complexes of such signs they designate various geometrical concepts, a circle, say, or an equilateral triangle. The complex sign (e.g., the drawn circle) exhibits what it is to be this or that figure, and it designates the (concept of) the figure itself. A primitive sign that is first seen as a part of one complex sign, as a radius of a circle, say, can later be seen as a part of another complex sign, say, as a side of a triangle.

24 Mechanical reasoning in Roman numeration Roman numerals function to record how many in a collection of things by using a stroke to stand in for one thing, then also abbreviations for larger collections: V for a collection of five, X for ten, L for fifty, and so on. A Roman numeral is thus a kind of a picture of a collection (through itself being a collection), one that can be mechanically manipulated to yield a solution to a problem, for instance, this: To divide three hundred and seventy-three by three.

25 Mechanical reasoning in Roman numeration We have a collection of three hundred and seventy-three: CCCLXXIII. The task is to divide, that is, literally (mechanically) separate, this collection into three separate collections. We have three C s and three I s so we begin our three collections with those: CI and CI and CI, leaving LXX. Now we rewrite LXX as XXXXXXX, seven X s. Putting two of those X s in each of our three collections gives us three collections of CXXI with X, or IIIIIIIIII remaining, which we again distribute among our collections as far as possible. The result is three collections of CXXIIII, with one I left over.

26 Mechanical reasoning in Hindu- Arabic numeration One can read a numeral of Hindu-Arabic numeration as one reads Roman numeration, each digit as standing for some one number of things with the position of the digit telling one what it is that one is counting in the collection. So considered, the complex sign 373 pictures a collection of three hundreds and seven tens and three units. It is read additively, as a Roman numeral is; and it can be manipulated mechanically as a Roman numeral is. This is how, and why, an adding machine works, by mechanically manipulating the counters in the various columns.

27 Mathematical reasoning in Hindu- Arabic numeration We saw that we could regard a drawn circle either mechanically, as depicting a state of affairs, or mathematically, as licensing inferences. So we can regard the numeral 373 either mechanically, as recording or picturing how many, or mathematically, as a complex sign for one number, as formulating what it is to be the number three hundred and seventy-three. On the second reading, only together in the complex sign do the digits function as numerals designating numbers. Independent of a context of use, the digits only express Fregean senses; they do not designate.

28 Mathematical reasoning in Hindu- Arabic numeration Because, on a mathematical reading, the numerals are complex signs for individual numbers, we can operate on those complex signs mathematically by applying rules. In this way we show things not about collections but about numbers say, that three hundred and seventy-three divided by three is one hundred and twenty-four with one remainder. Anyone watching, could take it that all one was doing was manipulating the signs mechanically, as one does with Roman numerals. But one could also be doing mathematics, treating the signs as displaying mathematical content in a way enabling one to reason in the system of signs.

29 What we have seen A mathematical language within which to reason has primitive signs that only express Fregean senses independent of a context of use (independent of, e.g., a use in a diagram or numeral). Whole complexes of those primitive signs serve to designate particular mathematical ideas, the notion of a circle, say, or that of the number three hundred and seventy-three. Because the signs are complex they can be manipulated according to rules to show the relationships of various mathematical ideas one to another.

30 Expressing the logical forms of mathematical concepts In a Euclidean diagram, geometrical form is displayed in drawn figures; one displays what it is to be, say, a circle for the purposes of diagrammatic reasoning. In equations of the formula language of arithmetic and algebra, computational form is displayed in such complex expressions as 373 and a 2 + b 2 what it is to be such things for the purposes of constructive, algebraic reasoning. What we are interested in now is the display of logical form, what it is to be, say, a prime number or to follow in a sequence for the purposes of deductive inference.

31 Examples of displays of the logical forms of concepts in Begriffsschrift Being a positive whole number:

32 Examples of displays of the logical forms of concepts in Begriffsschrift Being a positive whole number: (0 +1= )

33 Examples of displays of the logical forms of concepts in Begriffsschrift Being a positive whole number: (0 +1= ) Being a rational number:

34 Examples of displays of the logical forms of concepts in Begriffsschrift Being a positive whole number: (0 +1= ) Being a rational number: n (0 + = n ) (0 +1=n )

35 Examples of displays of the logical forms of concepts in Begriffsschrift Being a positive whole number: (0 +1= ) Being a rational number: n (0 + = n ) (0 +1=n ) Being a prime number:

36 Examples of displays of the logical forms of concepts in Begriffsschrift Being a positive whole number: (0 +1= ) Being a rational number: n (0 + = n ) (0 +1=n ) Being a prime number: d (0 + d = ) (1 +1=d ) d = (0 +1= )

37 Reasoning in Begriffsschrift In Part III of the 1879 logic Frege proves a theorem on the basis of four definitions. The proof, though strictly deductive, is ampliative, a real extension of our knowledge. Much as one does in diagrams in Euclid, and in equations in the symbolic language of arithmetic and algebra, one reasons in Frege s formula language of pure thought on the basis of the contents of concepts.

38 Frege s definitions Frege s definitions formulate the inferentially articulated contents of concepts. Although a definition can be merely truthfunctional, all the interesting ones contain inference licenses. To ascribe the concept in such a case is to issue an inference license that can be used to draw an inference provided that the condition of the inference has been met.

39 Frege s definitions Being hereditary in a sequence: d a F (a) f(d, a) F (d) F ( ) f(, ) This is a higher-order property of lower-order properties and relations. If a property F is hereditary in some sequence f then it may be inferred from the fact that the object d is F and the fact that f(d,a), that a is F.

40 Frege s definitions Following in a sequence: F a F(y) F(a) f(x, a) F( ) f(, ) f(x,y ) The basic idea here is that one can get to the following value y from the followed value x by repeated applications of the function f. There is no assumption that f is single-valued or forms a linear chain; it can branch, merge, and form rings.

41 Frege s definitions Belonging to a sequence: z x f(x,z ) f(x,z ) This definition does not contain an inference license but is merely truth-functional. To say that z belongs to the f-sequence beginning with x is merely to say that z either follows x in the f-sequence or is identical to x.

42 Frege s definitions Being a single-valued function: 88 e d a a e 9 9 >: >: f(d, a) >; f(d, e) I f(, ) >; To say that a function is single valued is to issue an inference license to the effect that if a and e are both the result of an application of f to d, then it can be inferred that a = e.

43 Theorem 133 f(m,y ) f(y,m ) f(x,y ) f(x,m ) I f(, ) If y and m both follow x in the f-sequence, f single valued, then either m follows y or y belongs to the f-sequence beginning with m.

44 Frege s proof of this theorem Frege proves theorem 133 in a completely rigorous way; every step in the proof conforms to his one rule of inference. But one quickly learns to omit obvious steps so as to focus on the important ones. We will look at some examples. But first we need a convention that will allow us to survey the whole course of reasoning.

45 Grounds and bridges All the steps in the proof have the form: given A-on-condition-that-B, and B, A is inferred. That is, the inference involves (1) a conditional premise and (2) a premise to the effect that the antecedent of the conditional is true. Premise (2) is what I will call the ground of the inference; the conditional premise (1) is the bridge carrying one from the ground to the conclusion.

46 Where a formula, say, axiom 1, plays the role of bridge taking one from a ground, say, axiom 2, to a conclusion, theorem 3, we display the inferential step thus: If we then construct theorem 4 on the basis of theorem 3 as ground, with 2 as bridge, we put: Using this convention the whole pattern of the derivation of theorem 133 from Frege s four definitions looks like this:

48 Examples of the general strategy We will consider two examples of the general strategy. In the first example, every inferential step, however trivial, will be included. In the second, trivial inferences will be omitted so as to focus on the significant steps, the joins.

49 F a F(y) F(a) f (x, a) δ F(α) α f (δ, α) γ f (x γ, y β ) β We begin with the definition of following in a sequence. The first thing we have to do is to transform that definition into a conditional.

50 a F(y) F(a) γ f (x γ, y β ) f β (x, a) δ F(α) α f (δ, α) γ f (x γ, y β ) β Now we need to reorder the conditions.

51 a F(y) γ f (x γ, y β ) β F(a) f (x, a) δ F(α) α f (δ, α) Now we reorganize according to the rule in axiom 2.

52 a F(y) γ f (x γ, y β ) β δ F(α) α f (δ, α) F(a) f (x, a) δ F(α) α f (δ, α) Now we reorganize according to the rule in axiom 2.

53 a a F(y) γ f (x γ, y β ) β δ F(α) α f (δ, α) F(a) f f (x, (x, a) a) δ F(α) α f f (δ, α) F(x) α) F(y) F(a) f (x, y) a) δ F(α) α f (δ, α) F(x) Notice what this theorem shows: that if some property F is hereditary, then it is hereditary The next step even in under Frege s weakening, presentation that is is, governed even if y is by not the rule the in result theorem of an 5 application and signals of that f to in x fact but we only are follows going in to the use f-sequence hypothetical after syllogism. x. We will do this directly. We assume as proven this theorem (derived ultimately from the definition of being hereditary in a sequence). But we need it in a slightly altered form. Now we can make our join.

55 To be shown: If y follows x, and z is the result of an application of f to y, then z follows x. f(x,z ) f(x,y ) f(y, z) This seems obviously true: if you can get to y from x by repeated applications of f, and z is the result of an application of f to y, then clearly you can get to z from x by repeated applications of f.

56 What we are trying to prove: f(x,z ) f(x,y ) f(y, z) The definitions we will need to prove it: F d a F (a) a f(d, a) F (d) F(y) F(a) f(x, a) F( ) f(, ) F ( ) f(, ) f(x,y )

57 F ( ) DANIELLE MACBETH f(x,z ) f(x,y ) f(y, z) Theorem 88 with concavity a F (a) f(, ) f(x, a) f(x,y ) f(y, z) F f(x,z,y ) F(z) F(y) F ( ) F F(z) F( ) f(, ) 3 with z for y F F a F(y) F(a) f(x, a) a F( ) a f(, ) F(a) f(, ) f(x, a) Theorem 89 f(x,z ) F(z) F ( ) F(a) f(x, a) f(x,y ) f(, ) F a F(a) a d a F (a) f(x, a) f(x,y ) f(y, z) f(x,y ) F(y) F(a) f(x, a) F ( ) f(d, a) F (d) F ( ) f(, )

58 F a F Theorem 87 F(y) F(a) f(x, a) f(, ) F ( ) F (y) 4. Various Theorems Derived in Theorem 84 f(, ) f(x, y) Theorem f(x,z 72 ) a F (a) f(x F (y) F ( ),y ) F (y) f(x, a) f(y, z) f(x f(x f(x, f(, y) ),y ),y ) F (x) F (x) f(y, z) Theorem 88 with concavity F ( ) F ( ) f(, ) f(, ) f(x F,z ) Theorem 73 with y for x and zf(z) for y Theorem 85 F(z) F( ) F (z) (y) F ( ) f(, ) F (y) f(y, f(x, z) y) a F(a) f(, ) F ( ) f(x, a) F ( ) F(a) f(, ) f(x f(, ) a,y ) a F( ) f(, ) f(x, a) Theorem 89 f(x,y ) F F (a) f(x, a) f(x,y ) a d a F (a) f(y, z) Theorem with y for x and z for f(x y,y ) F(y) F(a) f(x, a) F ( ) f(d, a) F (d) F (y) (x) F ( ) F (z) F ( ) f(, ) f(y, z) F ( ) f(, ) f(, )

59 What we have just seen From the definition of following in a sequence used twice (once to get a condition and once to get the conditioned judgment) and the definition of being hereditary used once (to get the other condition), we proved our little theorem in just two joins. All that was needed to discover this proof was attention to the particular structure of the theorem we aimed to prove, what was a condition and what was the conditioned content.

60 But... this strategy is not sufficient to account for the whole of the proof of theorem 133. The proofs of many theorems that are proven along the way are fully explicable by appeal to the general strategy we have just outlined. The proof as a whole is not so explicable. And that is what makes it interesting as a proof.

The plan Frege devises for the proof of theorem 133 It is trivial to prove this: What we are trying to prove: f(m,y ) (m y) f(x, y) f(x, m) I f(, ) f(y,m ) f(x,y ) f(x,m ) We can derive the one from

61 The plan Frege devises for the proof of theorem 133 It is trivial to prove this: What we are trying to prove: f(m,y ) (m y) f(x, y) f(x, m) I f(, ) f(y,m ) f(x,y ) f(x,m ) We can derive the one from the other provided that we do two things. We need to weaken these conditions, being the result of an application of f, to merely following in the f-sequence. But we can do that only if we weaken the identity of m and y to their merely being connected: m following y or y following m or m and y identical. I f(, )

62 The key idea of the proof First, prove of some property, here following x in the f-sequence, that it is hereditary: f(x,z ) f(y, z) f(x,y ) F (y) f(x,y f(x ),z ) F (x) f(y,z ) F ( ) f(x,y ) f(, ) Then use the definition of following in a sequence to derive a rule that licenses weakening of the condition that f(y,z). We have weakened the condition as required.

63 f(m,x ) f(x,m ) f(y,m ) f(y, x) I f(, ) f(x,m ) f(y,m ) f(y, x) I f(, ) (a x) f(y, a) f(y, x) I f(, ) f(x,a ) f(y, a) f(y, x) I f(, ) a f(x,m ) f(y,m ) f(x, a ) f(z,v ) f(y, v) f(z,y ) f(y, a)

64 To this point we have proven this: f(m,x ) What we need to prove is this: f(m,x ) f(x,m ) f(y, x) f(y,m ) I f(, ) f(x,m ) f(y,x ) f(y,m ) I f(, ) And we know what the strategy is: find some property that is hereditary and use that fact somehow to weaken the remaining condition. We have already shown that following and belonging are hereditary. What property is left? It is the property that Frege never so much as mentions, the property of being connected. Frege needs to show that connectedness is hereditary, and to do that he needs a second instance of the concept of connectedness. And here Frege does something nice. He constructs the needed instance of the concept inferentially: the required occurrence just pops up (in Manders technical sense) when the inference is made.

65 g(y) f(m,x ) h(y) f(x,y ) f(m,x ) f(x,m ) h(x) g( ) h( ) a f(x,m ) f(y, x) f(m,y ) f(, ) b c f(y,x ) f(y,m ) f(y,m ) d I f(, ) I f(, ) f(m,x ) f(m,x ) a c d f(x,m ) f(y, x) f(y,m ) f(x,m ) f(y, x) a b I f(, ) f(m,y )

66 f(m,x ) f(m,x ) f(x,m ) f(m,x ) f(y, x) f(x f(m,m,y) ) f(x,m ) f(m,x ) f(y,x ) f(x,m ) f(y,m ) f(y, x) f(y,m ) f(m I,,y ) ) f(y,m ) I f(x, ),m ) f(y, x) f(x,a ) I f( f(y, ),m ) f(y, a) f(y, x) I f(, ) f(x,m ) f(y, x) I f(, ) (a x) f(y,m ) f(y, a) f(y, x) I f(, ) a f(x, a ) f(y, a) f(z,v ) 88 e d a a e 9 9 f(y, v) f(z,y ) >: >: f(d, a) >; f(d, e) I f(, ) >;

67 133

Diagrammatic Reasoning in Frege s Begriffsschrift. Danielle Macbeth

Diagrammatic Reasoning in Frege s Begriffsschrift Danielle Macbeth In Part III of his 1879 logic 1 Frege proves a theorem in the theory of sequences on the basis of four definitions. He claims in Grundlagen