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8 How Does Mathematics Compare to Other Subjects? Other subjects: Study objects anchored in physical space and time. Acquire and prove knowledge primarily by inductive methods. Have subject specific concepts that do not (easily) apply to other subjects. Theories of the subconscious, id, and ego (psychology) cannot be used to explain the bonding of proteins (chemistry) Have concise definitions of their subject matter. Biology is the study of living organisms
9 Mathematics, On the Other Hand Studies objects that are seemingly independent of physical space and time. Numbers are not physical objects in nature. May acquire knowledge inductively, but proves knowledge by deductive methods. a 2 +b 2 =c 2 has been proven by logical deduction, not by observation. Has subject specific concepts that are easily and readily applicable in other fields of study Numbers are used to describe many phenomenon in science/non science fields. Has no concise definition Mathematics is? (This fact is really a major point Davis and Hersh are trying to make.)
10 What is mathematics? Some interesting quotes: A mathematician is a blind man in a dark room looking for a black cat, which isn t even there Charles Darwin Mathematics is the subject in which one never knows what he is talking about nor if what he says is true Bertrand Russell When the theories of mathematics are about reality, they are not certain; when they are certain, they are not about reality Albert Einstein
11 What is mathematics? Some interesting quotes: It is not of the essence of mathematics to be occupied with the ideas of number and quantity George Boole Mathematics is entirely free in its development, and its concepts are only linked by the necessity of being consistent Geoge Cantor Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician Friedrich Ludwig Gottfried Frege
12 What is mathematics? We can tentatively explore some definitions
13 Definition #1 Mathematics is: what mathematicians do. A similar definition has been used for Art: Art is that which an artist does. So what is the problem? Self Reference Circular Definition Not all activities of mathematicians are regarded as mathematics Mathematics is not exclusively practiced by mathematicians Does not clearly describe what it is to act as a mathematician. I think there is a germ of a good idea in this, however, that we ll develop over time in this course.
14 Definition #2 Mathematics is: Abstract Algebra Linear Algebra Euclidean Geometry Non Euclidean Geometry Fractal Geometry Differential Geometry Calculus Calculus of Variations Number Theory Logic & Set Theory Probability Statistics Topology Combinatorics Etc. The union of all these topics.
15 Definition #2 What is the problem with this definition? Verbose We expect a good definition to be concise and informative Unstable over time New mathematical subjects are always being created / discovered. A good definition should be capable of including future topics without having to rewrite the definition Emphasizes differences over similarities It does not answer what all these subjects have in common.
16 Definition #3 Mathematics is: Quantity Space Shape Change Chance Sets Groups Classes Rings Fields Functionals Operators Transformations Relations Etc The study of all these objects / ideas.
17 Definition #3 What is the problem with this definition? Same as before Not concise Unstable Emphasizes separation over unity, and, Begs the question of what are these objects? What do all these objects or ideas really have in common?
18 Definition #4 Examine this list Quantity Space Shape Change Chance Sets Groups Classes Rings Fields Functions Operators Connections Relations Etc. What does the study of all these things have in common? Logic is applied to all of them
19 Logicism Mathematics is: the class of all propositions of the form p implies q where p and q are propositions In other words: All mathematical objects can be defined solely with logic.
20 Logicism What is the problem with this definition? Not all logic is mathematics Law Crime Scene Investigation Not all mathematics can be defined by logic Russell (1902) found that logic alone is insufficient to deduce the basic laws of arithmetic. Godel (1931) found that there exist mathematical statements that cannot be proven (or disproven) via logical deductions. Logic is more properly considered a subtopic of mathematics Qualification as mathematics seems to be dependent upon the subject matter, not just method of reasoning.
21 Definition #5 Examine this list Quantity Space Shape Change Chance Sets Groups Classes Rings Fields Functions Operators Connections Relations Etc. What do all these things have in common? They are all abstract objects.
22 Platonism Mathematics is: The study of abstract objects, their properties, and interactions with other abstract objects. In other words: The unifying element of mathematics is not what math is, but rather where it is. By abstract, we mean separated from physical reality.
23 Platonism Example: When we solve the equation 2x 8 = 0 2x = 8 x = 4 Solutions seem to exist before they are written, and the equation seems to exist before it was thought of. But where, exactly, were they? In some alternate reality
24 Platonism So what s the problem? Assuming that mathematical objects are abstract Other objects are also abstract Ex. Freedom, Democracy, Power, Civilization, etc. Ex. Santa Claus, Tooth Fairy, Easter Bunny, Leprachaun Ex. La Chupacabra, Bigfoot, Los Angeles. Classification as abstract is an insufficient criterion to distinguish mathematical objects from non mathematical objects.
25 Platonism More problems Ontological Where do these objects really exist? Physical reality? Mental reality? Alternate reality? Spiritual reality? How long have they existed? If in an alternate reality, where exactly is this alternate reality? Epistemological How is it we have access to this alternate reality? How can we be sure of it? How do we know what exists in an alternate reality?
26 Definition #6 Examine this list once again Quantity Space Shape Change Chance Sets Groups Classes Rings Fields Functions Operators Connections Relations Etc. What does the study of all these things have in common? The use of symbols.
27 Formalism Mathematics is: A formal system of symbols, with rules of interaction between symbols. In other words: Mathematics is nothing more than meaningless marks on a piece of paper, written according to certain rules
28 Example: Formalism = 2 We normally consider this a fundamental truth But according to formalism, this is an arbitrary truth. We could just as easily say that = 3 And this is equally as valid, because the meaning of the symbols is arbitrary The symbols 1, +, =, and 3 can mean whatever we want them to mean. Essentially one can define any rule for the interaction between these symbols
29 Formalism What s the problem? The basic object of mathematics is a set. A set of objects is a well defined collection of ANYTHING Meaningless has no meaning Arbitrary could mean anything Mathematics is not arbitrary and not meaningless. Symbols are used in mathematics, but math is more than just symbols. In practice, mathematics is never reduced to just symbols.
30 A Good Definition of Mathematics Should be: Concise Informative And should also account for: Use of logic Abstraction Symbolism Unreasonable effectiveness Self consistency How mathematicians actually act when they do it
31 Is There No definition? It is likely that the phrase everything that is mathematics does not form a well defined set. Many mathematicians: Believe a definition exists. Insist on precise definition of mathematical terms, but do not insist on a precise definition of mathematics itself. Tend to side with Platonism, but when hard pressed to provide a definition, tend to resort to formalism (Hersh, 1999)
32 Infallibility: Other Characteristics of Mathematics
33 Infallibility Once a piece of mathematics is discovered and proved, it is true and remains true forever.
34 Davis and Hersh in Context Davis and Hersh s The Mathematical Experience is meant to help the reader appreciate the experience of being a mathematician, with all its joys, frustrations, dilemmas, the thrill of victory, the agony of defeat. It does a good job of that, and for that reason is a worthy book to read. But also:
35 Davis and Hersh in Context Davis and Hersh can be read as an argument against all of the above definitions and characteristics, except perhaps the germ of a good idea I alluded to in #1. This includes: The union of all the subtopics Platonism Formalism Logicism Fallibility
36 Davis and Hersh in Context Moreover, they use their strongest arguments, in some sense, against the beliefs that most mathematicians seem to hold about the nature of mathematics. This is perhaps most evident in The Ideal Mathematician but is evident elsewhere as well.
37 Davis and Hersh in Context Chapter 1: Ulam s Dilemma: There are far too many pieces of mathematics for one person to know and by implication, too many sub areas of specialization to serve as a basis for a useful definition of mathematics.
38 Davis and Hersh in Context: Pages 32 33: Chapter 2 What was in Archimedes head was different from what was in Newton s head; they had to understand the Pythagorean Theorem differently. Does this have implications for Platonism? Does this have implications for Formalism? How about Logicism?
39 Davis and Hersh in Context: Ideal Mathematician: Chapter 2 truths which are valid forever, from the beginning of time, even in the most remote corner of the universe. to read his proofs, one must be privy to the whole subculture of motivations, standard arguments and examples, habits of thought, and agreed upon modes of reasoning.
40 Davis and Hersh in Context: Ideal Mathematician: Chapter 2 There is no way we could convince a selfconfident skeptic that the things we are talking about make sense, let alone exist.
41 Davis and Hersh in Context: Chapter 2 Physicist: If the physicist uses mathematics, then what are the implications of: truth in mathematics... is reasoning that leads to correct physical relationships. proof is for cosmetic purposes...
42 Davis and Hersh in Context: Pages 59 60: Chapter 2 In the final analysis, there can be no formalization of what is right and how we know it is right, what is accepted, and what the mechanism for acceptance is.
43 Everyone Sing Along! You can ask some interesting question about these definitions and how they fit with what we know about history. For example, What position is strengthened by the evidence that almost everyone came up with Pascal s Triangle, at different times and in different cultures and places? Another Example: Lobachevski, Bolyai, and Gauss
44 What Davis and Hersh Leads to: Point 1 is that mathematics is a social historic reality. This is not controversial. All that Platonists, formalists, intuitionists, and others can say against it is that it s irrelevant to their concept of philosophy. Point 2 is controversial: There s no need to look for a hidden meaning or defintion of mathematics beyond its social historic cultural meaning. Socialhistoric is all it needs to be. Forget foundations, forget immaterial, inhuman reality. Reuben Hersh, What is Mathematics, Really?, p. 23.
45 By the way, I unabashedly borrowed and revised a lot of this slide presentation from: o2/math1319/philosophy/philosophy.ppt
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