Guide Philosophy of Mathematics: An Introduction to a World of Proofs and Pictures

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Mathematics is the art of giving the same name to different things. Mathematics is one of the essential emanations of the human spirit — a think to be valued in and for itself like art or poetry. Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost.

Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere. Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self-consciousness. The significance of mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it informs us of much that depends on our minds.

Mathematics is the supreme judge; from its decisions there is no appeal. Mathematics is no more computation than typing is literature. Mathematics is the language with which God wrote the universe. Mathematics is the handwriting on the human consciousness of the very spirit of life itself. Mathematics is the queen of science.

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. Mathematics consists in proving the most obvious thing in the least obvious way. Mathematics is like love; a simple idea, but it can get complicated. Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state. What you do is trial and error, experimentation, guesswork.

Mathematics is written for mathematicians. Mathematics is a great motivator for all humans..

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Because its career starts with zero and it never end infinity. Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field. Mathematics is not only real, but it is the only reality. Mathematics is often erroneously referred to as the science of common sense. Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper. Ma thematics is an independent world created out of pure intelligence.

Mathematics is, as it were, a sensuous logic, and relates to philosophy as do the arts, music, and plastic art to poetry. Mathematics is the science which uses easy words for hard ideas. The model theoretic approach shows how different logics interact with different mathematical structures. Mortensen has followed through on this in a wide array of subjects, from the differential calculus to vector spaces to topology to category theory, always asking: Under what conditions is identity well-behaved? It turns out that the answer to this question is extremely sensitive to small changes in logic and interpretations, and the answer can often be "no.

Most of the results obtained to date have been through the model theoretic approach, which has the advantage of maintaining a connection with classical mathematics. The model theory approach has the same disadvantage, since it is unlikely that radically new or robustly inconsistent ideas will arise from always beginning at classical ideas. It is often thought that inconsistent mathematics faces a grave problem. A very common mathematical proof technique is reductio ad absurdum. The concern, then, is that if contradictions are not absurd—a fortiori, if a theory has contradictions in it—then reductio is not possible.

How can mathematics be done without the most common sort of indirect proof? The key to working inconsistent mathematics is its logic.

Philosophy of Mathematics: An Introduction to a World of Proofs and Pictures - CRC Press Book

Much hinges on which paraconsistent logic we are using. For instance, in da Costa's systems, if a proposition is marked as "consistent," then reductio is allowed.

Similarly, in most relevance logics, contraposition holds. And so forth. The reader is recommended to the bibliography for information on paraconsistent logic. Independently of logic, the following may help. In classical logic, all contradictions are absurd; in a paraconsistent logic this is not so.

But some things are absurd nevertheless. Classically, contradiction and absurdity play the same role, of being a rejection device, a reason to rule out some possibility. In inconsistent mathematics, there are still rejection devices. Anything that leads to a trivial theory is to be rejected. Now, we are looking for interesting inconsistent structure.

There are many consistent structures that mathematicians do not, and will never, investigate, not by force of pure logic but because they are not interesting. Inconsistent mathematicians, irrespective of formal proof procedures, do the same. Intuitively, M. Escher's "Ascending, Descending" is a picture of an impossible structure—a staircase that, if you walked continuously along it, you would be going both up and down at the same time.

Such a staircase may be called impossible.

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Nevertheless, the picture is coherent and interesting. What sorts of mathematical properties does it have? The answers to this and more would be the start of an inconsistent geometry. So far, the study has focused on the impossible pictures themselves. A systematic study of these pictures is being carried out by the Adelaide school. Two main results have been obtained. First, Bruno Ernst conjectured that one cannot rotate an impossible picture. This was refuted in by Mortensen; later, Quigley designed computer simulations of rotating impossible Necker cubes.

It is thought that these forms exhaust the universe of impossible pictures. If so, an important step towards a fuller geometry will have been taken, since, for example, a central theme in surface geometry is to classify surfaces as either convex, flat, or concave. Most recently, Mortensen and Leishman have characterized Necker cubes, including chains of Neckers, using linear algebra. Otherwise, algebraic and analytic methods have not yet been applied in the same way they have been in classical geometry.

Inconsistent equational expressions are not at the point where a robust answer can be given to questions of length, area, volume etc. On the other hand, as the Adelaide school is showing, the ancient Greeks do not have a monopoly on basic "circles drawn in sand" geometric discoveries.

Set theory is one of the most investigated areas in inconsistent mathematics, perhaps because there is the most consensus that the theories under study might be true. It is here we have perhaps the most important theorem for inconsistent mathematics, Ross Brady's proof that inconsistent set theory is non-trivial. These are the principles of comprehension a. In symbols,. Again, these assumptions seem true. When the first assumption, the principle of comprehension, was proved to have inconsistent consequences, this was felt to be highly paradoxical.

The inconsistent mathematician asserts that a theory implying an inconsistency is not automatically equivalent to a theory being wrong. Quine's new foundations. In this system, axioms like those of standard set theory are assumed, along with the existence of a Russell set. This shows that some sets are self-membered. On the other hand, in perhaps the first truly combinatorial theorem of inconsistent mathematics, Arruda and Batens proved. This says that every set is a member of a non-self-membered set.

The Arruda-Batens result was obtained with a very weak logic, and shows that there are real set theoretical theorems to be learned about inconsistent objects. Arruda further showed that. Routley, meanwhile, in took up his own dialetheic logic and used it on a full comprehension principle. Routley went as far as to allow a comprehension principle where the set being defined could appear in its own definition. A more mundane example of a set appearing in its own defining condition could be the set of "critics who only criticize each other. Routley indicated that the usual axioms of classical set theory can be proven as theorems—including a version of the axiom of choice—and began work towards a full reconstruction of Cantorian set theory.

The crucial step in the development of Routley's set theory came in when Brady adapted an idea from to produce a model of dialetheic set theory, showing that it is not trivial. Brady proves that there is a model in which all the axioms and consequences of set theory are true, including some contradictions like Russell's, but in which some sentences are not true. By the soundness of the semantics, then, some sentences are not provable, and the theory is decidedly paraconsistent.

What are Numbers? Philosophy of Mathematics

A stream of papers considering models for paraconsistent set theory has been coming out of Europe as well. Olivier Esser has determined under what conditions the axiom of choice is true, for example. See Hinnion and Libert for an opening into this work. Inconsistent set theory, on the other hand, appears to be able to answer some of these questions.

The existence of large cardinals is undecidable by classical set theory. But recall the universe, as we did in the introduction section 1 , and its size V. Almost obviously, V is such large a cardinal, just because everything is smaller than it. Taking the full sweep of sets into account, the hypothesis is true.

Set theory is the lingua franca of mathematics and the home of mathematical study of infinity. Since Zeno's paradoxes it has been obvious that there is something paradoxical about infinity. Since Russell's paradox, it has been obvious that there is something paradoxical about set theory. So a rigorously developed paraconsistent set theory serves two purposes. First, it provides a reliable inconsistent foundation for mathematics, at least in the sense of providing the basic toolkit for expressing mathematical ideas. Second, the mathematics of infinity can be refined to cover the inconsistent cases like Cantor's paradox, and cases that have yet to be considered.

See the references for what has been done in inconsistent set theory so far; what can be still be done in remains one of the discipline's most exciting open questions. An inconsistent arithmetic may be considered an alternative or variant on the standard theory, like a non-euclidean geometry. Like set theory, though, there are some who think that an inconsistent arithmetic may be true, for the following reason.

This means that any consistent theory of numbers will always be an incomplete fragment of the whole truth about numbers. Priest has argued in a series of papers that this means that the whole truth about numbers is inconsistent. The standard axioms of arithmetic are Peano's, and their consequences—the standard theory of arithmetic—is called P A.

N is a model of arithmetic because it makes all the right sentences true. In Skolem noticed that there are other consistent models that make all the same sentences true, but have a different shape—namely, the non-standard models include blocks of objects after all the standard members of N.

The consistent non-standard models are all extensions of the standard model, models containing extra objects. Inconsistent models of arithmetic are the natural dual, where the standard model is itself an extension of a more basic structure, which also makes all the right sentences true. Part of this idea goes back to C. An important discovery in the late 19th century was that arithmetic facts are reducible to facts about a successor relation starting from a base element.

In modular arithmetic, a successor function is wrapped around itself. Gauss no doubt saw this as a useful technical device. Inconsistent number theorists have considered taking such congruences much more seriously. There he took the paraconsistent logic R and added to it axioms governing successor, addition, multiplication, and induction, giving the system R.

In Meyer proved that his arithemtic is non-trivial, because R has models. Since relevance logicians have studied the relationship between R and PA. Their hope was that R contains PA as a subtheory and could replace PA as a stronger, more genuine arithmetic. The outcome of that project for our purposes is the development of inconsistent models of arithmetic. Priest has found inconsistent arithmetic to have an elegant general structure. Rather than describe the details, here is an intuitive example. We imagine the standard model of arithmetic, up to an inconsistent element.

This n is suspected to be a very, very large number, "without physical reality or psychological meaning. Any fact true of numbers greater than n are true of n , too, because after n , all numbers are identical to n. No facts from the consistent model are lost. This technique gives a collapsed model of arithmetic. Let T be all the sentences in the language of arithmetic that are true of N; then let T n similarly be all the sentences true of the numbers up to n , an inconsistent number theory.

The sentences of T n are representable in T n , and its language contains a truth predicate for T n. The theory can prove itself sound. Yet as Meyer proved, the non-triviality of T n can be established in T n by a finite procedure. This means that T n is decidable, and that there must be axioms guaranteed to deliver every truth about the collapsed model. This means that an inconsistent arithmetic is coherent and complete. Newton and Leibniz independently developed the calculus in the 17th century. They presented ingenious solutions to outstanding problems rates of change, areas under curves using infinitesimally small quantities.

Consider a curve and a tangent to the curve. Where the tangent line and the curve intersect can be though of as a point. If the curve is the trajectory of some object in motion, this point is an instant of change. But a bit of thought shows that it must be a little more than a point—otherwise, as a measure a rate of change, there would be no change at all, any more than a photograph is in motion. There must be some smudge. An infinitesimal would respect both these concerns, and with these provided, a circle could be construed as infinitely many infinitesimal tangent segments.

Infinitesimals were essential, not only for building up the conceptual steps to inventing calculus, but in getting the right answers. Yet it was pointed out, most famously by Bishop George Berkeley , that infinitesimals were poorly understood and were being used inconsistently in equations. Calculus in its original form was outright inconsistent. Here is an example. Using the original definition of a derivative,.

Philosophy of Mathematics

It marks a small but non-trivial neighborhood around x , and can be divided by, so it is not zero. This example suggests that paraconsistent logic is more than a useful technical device. The example shows that Leibniz was reasoning with contradictory information, and yet did not infer everything.

On the contrary, he got the right answer. Nor is this an isolated incident. Mathematicians seem able to sort through "noise" and derive interesting truths, even out of contradictory data sets. To capture this, Brown and Priest have developed a method they call "chunk and permeate" to model reasoning in the early calculus. Each chunk is consistent, without conflicting information, and one can reason using classical logic inside of a chunk.

Then a permeation relation is defined which controls the information flow between chunks. Brown and Priest propose this as a model, or rational reconstruction, of what Newton and Leibniz were doing. Another, more direct tack for inconsistent mathematics is to work with infinitesimal numbers themselves. There are classical theories of infinitesimals due to Abraham Robinson the hyperreals , and J. Conway the surreals. Mortensen has worked with differential equations using hyperreals.

Another approach is from category theory. The category theory approach is the most like inconsistent mathematics, then, since it involves a change in the logic. In general the concept of continuity is rich for inconsistent developments. Is there a computer program to decide, for any arithmetic statement, whether or not the statement can be proven? Is there a program to decide, for any arithmetic statement, whether or not the statement is true? It is natural to extend these ideas into computer science. Hilbert's program demands certain algorithms —a step-by-step procedure that can be carried out without insight or creativity.

Is there a program E that can tell us in advance whether a given program will halt or not? A paraconsistent system can occasionally produce contradictions as an output, while its procedure remains completely deterministic. It is not that the machine occasionally does and does not produce an output. There is, in principle, no reason a decision program cannot exist.

Richard Sylvan identifies as a central idea of paraconsistent computability theory the development of machines "to compute diagonal functions that are classically regarded as uncomputable. Important results have been obtained by the paraconsistent school in Brazil—da Costa and Doria in , and Agudelo and Carnielli in Like quantum computation, though, at present the theory of paraconsistent machines outstrips the hardware.

Machines that can compute more than Turing machines await advances in physics. Priest's In Contradiction is the best place to start. The second edition contains material on set theory, continuity, and inconsistent arithmetic summarizing material previously published in papers.


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A critique of inconsistent arithmetic is in Shapiro Franz Berto's book, How to Sell a Contradiction , is harder to find, but also an excellent and perhaps more gentle introduction. Some of da Costa's paraconsistent mathematics is summarized in the interesting collection Frontiers of Paraconsistency the proceedings of a world congress on paraconsistency edited by Batens et al. More details are in Jacquette's Philosophy of Logic handbook; Beall's paper in that volume covers issues about truth and inconsistency.

Those wanting more advanced mathematical topics should consult Mortensen's Inconsistent Mathematics For impossible geometry, his recent pair of papers with Leishman are a promising advance.