Oct 13, 2012

The System of Mathematics: Peterson’s and Hempel’s Notions

By Marsigit
Yogyakarta State University

Peterson, I., 1998, elaborated that at the beginning of the 20th century, the great German mathematician David Hilbert (1862-1943) advocated an ambitious program to formulate a system of axioms and rules of

inference that would encompass all mathematics, from basic arithmetic to advanced calculus; his dream was to codify the methods of mathematical reasoning and put them within a single framework. Hilbert1 insisted that such a formal system of axioms and rules should be consistent, meaning that one can't prove an assertion and its opposite at the same time; he also wanted a scheme that is complete, meaning that one can always prove a given assertion either true or false. Hilbert2 argued that there had to be a clear-cut mechanical procedure for deciding whether a certain proposition follows from a given set of axioms; hence, given a well-defined system of axioms and appropriate rules of inference, it would be possible, though not actually practical, to run through all possible propositions, starting with the shortest sequences of symbols, and to check which ones are valid. In principle, such a decision procedure would automatically generate all possible theorems in mathematics.3

On the other hand, it was elaborated that formal mathematics builds on formal logic; 4it reduces mathematical relationships to questions of set membership; the only undefined primitive object in formal mathematics is the empty set that contains nothing at all. It was claimed that almost every mathematical abstraction that has ever been investigated can be derived as a set of the axioms of set theory and almost every mathematical proof ever constructed can be made assuming nothing beyond those axioms. 5It was also stated that if infinity is a potential and never a completed reality then infinite sets do not exist; therefore, mathematicians try to define the most general infinite structures imaginable because that seems to give the most bang for the buck; if no infinite sets exist this would be the construction of a fantasy. Further, 6it was claimed that mathematics should be directly connected to properties of nondeterministic programs in a potentially infinite universe; this would limit extensions to a fragment of the countable ordinals and the sets that can be constructed from them. The objects definable within a formal mathematical system no matter what axioms of infinity it includes are countable; and that a formal system can be interpreted as a computer program for generating theorems in which such a program can output all of the names of the objects or sets definable with the system. Further, all real numbers and for that matter larger cardinals that can ever be defined in any mathematical system that finite creatures create will be countable; they will not be countable from within the system.7

Peterson, I., 1998, noted that what Hilbert was saying is that we can solve a problem if we are clever enough and work at it long enough; and mathematician Gregory J. Chaitin and Thomas J. Watson didn't believe that in principle there was any limit to what mathematics could achieve. However, in the 1930s, 8Kurt Godel (1906-1978) proved that no such decision procedure is possible for any system of logic made up of axioms and propositions sufficiently sophisticated to encompass the kinds of problems that mathematicians work on every day; he showed that if we assume that the mathematical system is consistent, then we can show that it is incomplete. 9Peterson said that in Godel's realm, no matter what the system of axioms or rules is, there will always be some assertion that can be neither proved nor invalidated within the system; indeed, mathematics is full of conjectures assertions awaiting proof with no assurance that definitive answers even exist.

Chaitin10 proved that no program can generate a number more complex than itself; in other words, a 1-pound theory can no more produce a 10-pound theorem than a 100-pound pregnant woman can birth a 200-pound child. Conversely, Chaitin11 also showed that it is impossible for a program to prove that a number more complex than the program is random; hence, to the extent that the human mind is a kind of computer, there may be a type of complexity so deep and subtle that our intellect could never grasp it; whatever order may lie in the depths would be inaccessible, and it would always appear to us as randomness. At the same time, proving that a sequence is random also presents insurmountable difficulties; there's no way to be sure that we haven't overlooked a hint of order that would allow even a small compression in the computer program that produces the sequence. Peterson, I., 1998, stated that Chaitin's result suggests that we are far more likely to find randomness than order within certain domains of mathematics; indeed, his complexity version of Godel's theorem states that although almost all numbers are random, there is no formal axiomatic system that will allow us to prove this fact.

Further, Peterson, I., 1998, concluded that Chaitin's work suggests that there is an infinite number of mathematical statements that one can make about, say, arithmetic that can't be reduced to the axioms of arithmetic, so there's no way to prove whether the statements are true or false by using arithmetic; in Chaitin's view, that's practically the same as saying that the structure of arithmetic is random. Chaitin12 summed up that they are mathematical facts which are analogous to the outcome of a coin toss and we can never actually prove logically whether they are true; he added that in the same way that it is impossible to predict the exact moment at which an individual atom undergoes radioactive decay, mathematics is powerless to answer particular questions; while physicists can still make reliable predictions about averages over large ensembles of atoms, mathematicians may in some cases be limited to a similar approach; that makes mathematics much more of an experimental science than many mathematicians would be willing to admit.

Hempel, C.G., 2001, argued that any self-consistent postulate system of mathematics admits, nevertheless, many different interpretations of the primitive terms , whereas a set of definitions in the strict sense of the word determines the meanings of the definienda in a unique fashion. The much broader system of Peano’s postulate thus obtained is still incomplete in the sense that not every number in it has a square root, and more generally, not every algebraic equation whose coefficients are all numbers of the system has a solution in the system; this suggests further expansions of the number system by the introduction of real and finally of complex numbers. Hempel13 concluded that on the basis of this postulate the various arithmetical and algebraic operations can be defined for the numbers of the new system, the concepts of function, of limit, of derivative and integral can be introduced, and the familiar theorems pertaining to these concepts can be proved, so that finally the huge system of mathematics as here delimited rests on the narrow basis of Peano's system; every concept of mathematics can be defined by means of Peano's three primitives, and every proposition of mathematics can be deduced from the five postulates enriched by the definitions of the non-primitive terms; these deductions can be carried out, in most cases, by means of nothing more than the principles of formal logic; the proof of some theorems concerning real numbers, however, requires one assumption which is not usually included among the latter and this is the so-called axiom of choice in which it asserts that given a class of mutually exclusive classes, none of which is empty, there exists at least one class which has exactly one element in common with each of the given classes.

Hempel, C.G., 2001, claimed that by virtue of this principle and the rules of formal logic, the content of all of mathematics can thus be derived from Peano's modest system i.e. a remarkable achievement in systematising the content of mathematics and clarifying the foundations of its validity. According to him, the Peano system permits of many different interpretations, whereas in everyday as well as in scientific language, we attach one specific meaning to the concepts of arithmetic. Hempel14 insisted that if therefore mathematics is to be a correct theory of the mathematical concepts in their intended meaning, it is not sufficient for its validation to have shown that the entire system is derivable from the Peano postulates plus suitable definitions; rather, we have to inquire further whether the Peano postulates are actually true when the primitives are understood in their customary meaning. If the definitions here characterized are carefully written out i.e. this is one of the cases where the techniques of symbolic, or mathematical, logic prove indispensable, it is seen that the definiens of every one of them contains exclusively terms from the field of pure logic. In fact, it is possible to state the customary interpretation of Peano's' primitives, and thus also the meaning of every concept definable by means of them, and that includes every concept of mathematics:

-- in terms of the following seven expressions (in addition to variables such as "x" and "C"): not, and, if -- then; for every object x it is the case that . . .; there is some object x such that . . .; x is an element of class C; the class of all things x such that . . . And it is even possible to reduce the number of logical concepts needed to a mere four: The first three of the concepts just mentioned are all definable in terms of "neither--nor," and the fifth is definable by means of the fourth and "neither--nor." Thus, all the concepts of mathematics prove definable in terms of four concepts of pure logic.The definition of one of the more complex concepts of mathematics in terms of the four primitives just mentioned may well fill hundreds or even thousands of pages; but clearly this affects in no way the theoretical importance of the result just obtained; it does, however, show the great convenience and indeed practical indispensability for mathematics of having a large system of highly complex defined concepts available. 15

Hempel, C.G., 2001, notified that a stable self-contained system of basic principles is the distinctive feature of mathematical theories; a mathematical model of some natural process or a technical device is essentially a stable model that can be investigated independently of its "original" and, thus, the similarity of the model and the "original" is only a limited one; only such models can be investigated by mathematicians. Hempel16 thought that any attempt to refine the model i.e. to change its definition in order to obtain more similarity with the "original", leads to a new model that must remain stable again, to enable a mathematical investigation of it; hence, mathematical theories are the part of science we could continue to do if we woke up tomorrow and discovered the universe was gone. Hempel17 argued that a mathematical model, because it is stable, is not bound firmly to its "original" source; it may appear that some model is constructed badly, in the sense of the correspondence to their "original" source, yet its mathematical investigation goes on successfully. According to him, since a mathematical model is defined very precisely, it "does not need" its "original" source any more. One can change some model or obtaining a new model not only for the sake of the correspondence to the "original" source, but also for a mere experiment. In this way people may obtain easily various models that do not have any real "originals", an even entire branches of mathematics have been developed that do not have and cannot have any applications to real problems. Hempel18 claimed that the stable self-contained character of mathematical models makes such "mutations" possible and even inevitable and no other branch of science knows such problems.

Hempel, C.G., 2001, noted that, in mathematics, theorems of any theory consist, as a rule, of two parts - the premise and the conclusion; therefore, the conclusion of a theorem is derived not only from a fixed set of axioms, but also from a premise that is specific to this particular theorem; and this premise is it not an extension of the fixed system of principles. He perceived that Mathematical theories are open for new notions; thus, in the Calculus after the notion of continuity the following connected notions were introduced: break points, uniform continuity, Lipschitz's conditions, etc. and all this does not contradict the thesis about the fixed character of principles axioms and rules of inference, yet it does not allow "working mathematicians" to regard mathematical theories as fixed ones. On the other hand, 19Burris, 1997, believed that the science of mathematics finds itself necessarily and essentially based on abstraction, since mathematicians never empirically test, for example, putting 200 mice and 200 mice to create 400 mice; the fact that all of mathematics is abstract was one of the motives of intuitionists to think mathematics is a sole product of the mind; however, their inference that because we do not empirically prove a mathematical statement, it is only a figment of imagination, was wrong. While 20Shapiro, S., 2000, indicated that Mathematical logic and, in particular, model theory can perhaps be construed as a theory that quantifies over structures; to say that a class of sentences is satisfiable may be to say that there is a structure that satisfies it. It is more common to construe the variables of model theory as ranging over sets. According to him, sets are currently taken to be places within the set-theoretical-hierarchy-structure. Thus, like any other branch of mathematics, model theory is seen as the study of a particular structure; the thesis that mathematics can be reduced to set theory amounts to a claim that any given mathematical structure except that of set theory itself can be modeled in the set-theoretic structure and, moreover, that the latter captures the relevant relationships between structures.21

1 In Peterson, I., 1998, The Limits of Mathematics, The Mathematical Association of America, http://www.sciencenews.org/)
3 Ibid.
4 In -------, 2004,Formal mathematics, Mountain Math Software, webmaster@mtnmath.com
5 Ibid.
6 Ibid.
7 Ibid.
8 In Peterson, I., 1998, The Limits of Mathematics, The Mathematical Association of America,9 http://www.sciencenews.org/)
10 Ibid.
14Hempel, C.G., 2001, On the Nature of Mathematical Truth, http://www.ltn.lv/ ~podniek/gt.htm
19 Ibid.
20 Ibid.
21 Ibid


  1. Nama : Irna K.S.Blegur
    Nim : 16709251064
    kelas : PM D 2016(PPS)
    Pandangan Peterson mengenai Sistem Matematika:
    1. Hilbert menegaskan bahwa suatu sistem formal dari aksioma dan aturan harus konsisten, yang berarti bahwa seseorang tidak dapat membuktikan sebuah pernyataan dan kebalikannya pada saat yang sama, ia juga menginginkan skema yang lengkap, artinya satu selalu dapat membuktikan pernyataan yang diberikan bisa benar atau salah.
    2. Hilbert berpendapat bahwa harus ada prosedur yang jelas untuk memutuskan apakah suatu proposisi tertentu berikut dari himpunan aksioma, dengan itu, diberikan sebuah sistem yang jelas dari aksioma dan aturan inferensi yang tepat, akan lebih mungkin, meskipun tidak benar-benar praktis, untuk menjalankan melalui semua proposisi mungkin, dimulai dengan urutan terpendek simbol, dan untuk memeriksa mana yang valid. Pada prinsipnya, suatu prosedur keputusan secara otomatis akan menghasilkan semua teorema mungkin dalam matematika.
    3Ia menjelaskan bahwa matematika formal didasarkan pada logika formal
    4. mengurangi hubungan matematis untuk pertanyaan keanggotaan himpunan; objek primitif hanya terdefinisi dalam matematika formal adalah himpunan kosong yang berisi apa-apa. Ada klaim bahwa hampir setiap abstraksi matematika yang pernah diselidiki dapat diturunkan sebagai seperangkat aksioma teori himpunan dan hampir setiap bukti matematis yang pernah dibangun dapat dibuat dengan asumsi tidak ada di luar yang aksioma.
    5.Itu juga menyatakan bahwa jika tak terhingga merupakan potensi dan tidak pernah menjadi kenyataan selesai maka himpunan terbatas tidak ada, karena itu, ahli matematika mencoba untuk mendefinisikan struktur tak terbatas yang paling umum dibayangkan karena itu tampaknya memberikan harapan paling baik, jika himpunan tidak terbatas ada maka akan menjadi landasan matematika yang kokoh.
    6.Lebih lanjut, ia menyatakan bahwa matematika harus langsung terhubung ke sifat program non-deterministic di alam semesta yang potensial tidak terbatas, hal ini akan membatasi ekstensi untuk sebuah himpunan bilangan ordinal dan himpunan yang dapat dibangun dari mereka. Obyek didefinisikan dalam suatu sistem matematis yang formal tidak peduli apakah aksioma tak terhingga itu termasuk yang dimasukkan, dan bahwa sistem formal dapat diartikan sebagai suatu program komputer untuk menghasilkan teorema di mana program tersebut dapat menghasilkan semua nama-nama benda atau himpunan yang didefinisikan dalam sistem tersebut.
    7. Selanjutnya, semua bilangan kardinal yang lebih besar yang pernah didefinisikan dalam sistem matematika yang terbatas, tidak akan dihitung dari dalam sistem tersebut.

  2. 8.Kurt Godel (1906-1978) membuktikan bahwa tidak ada prosedur keputusan tersebut adalah mungkin untuk setiap sistem logika yang terdiri dari aksioma dan proposisi cukup canggih untuk mencakup jenis masalah matematika yang hebat yang bekerja pada setiap hari; ia menunjukkan bahwa jika kita asumsikan bahwa sistem matematika konsisten, maka kita bisa menunjukkan bahwa itu tidak lengkap.
    9.Dalam pikiran Godel, tidak peduli apa sistem aksioma atau aturannya, akan selalu ada beberapa pernyataan yang dapat tidak terbukti atau tidak valid dalam sistem. Memang, matematika penuh dengan pernyataan dugaan dan menunggu bukti dengan jaminan bahwa jawaban tertentu telah pernah ada.
    10.Chaitin membuktikan bahwa suatu prosedur tidak dapat menghasilkan hasil yang lebih kompleks dari pada prosedur itu sendiri, dengan kata lain, dia membuat teori bahwa wanita berbobot 1-pon tidak bisa melahirkan bayi berbobot 10-pon. Wanita berbobot 10 pon tidak bisa melahirkan bayi 100 pon, dst.
    11.Chaitin juga menunjukkan bahwa tidak mungkin membuat prosedur untuk membuktikan bahwa sejumlah kompleksitas bersifat acak, maka, sejauh bahwa pikiran manusia adalah sejenis komputer, mungkin ada jenis kompleksitas begitu mendalam dan halus yang akal kita tidak pernah bisa memahami nya; urutan apapun yang mungkin terletak pada kedalaman akan dapat diakses, dan selalu akan muncul untuk kita sebagai keacakan. Pada saat yang sama, membuktikan bahwa berurutan adalah acak juga dapat mengatasi kesulitan, tidak ada cara untuk memastikan bahwa kita tidak diabaikan. Peterson, I., 1998, menyatakan bahwa hasil Chaitin ini menunjukkan bahwa kita jauh lebih mungkin untuk menemukan keacakan dari ketertiban dalam domain matematika tertentu; kompleksitas versin teorema Godel menyatakan bahwa meskipun hampir semua bilangan adalah acak, tidak ada sistem formal aksiomatis yang akan memungkinkan kita untuk membuktikan fakta ini.
    12. Chaitin menyimpulkan bahwa struktur matematika adalah fakta matematis yang analog dengan hasil dari sebuah lemparan koin dan kita tidak pernah bisa benar-benar membuktikan secara logis apakah itu adalah benar, ia menambahkan bahwa dengan cara yang sama bahwa tidak mungkin untuk memprediksi saat yang tepat di mana seorang individu yang terkena radiasi atom mengalami peluruhan radioaktif. Matematika tak berdaya untuk menjawab pertanyaan tertentu, sedangkan fisikawan masih dapat membuat prediksi yang dapat diandalkan tentang rata-rata lebih dari besar dari atom, ahli matematika mungkin dalam beberapa kasus terbatas pada pendekatan yang sama; yang membuat matematika jauh lebih dari ilmu pengetahuan eksperimental.

  3. Saepul Watan
    S2 P.Mat Kelas C 2016

    Bismilahir rahmaanir rahiim..
    Assalamualaikum wr..wb...

    Artikel The System of Mathematics: Peterson’s and Hempel’s Notions memaparkan pandangan-pandangan dari Peterson dan Hempel tentang pembuktian dengan aksioma dalam teorema matematika. Teorema-teorema dalam matematika dapat dibuktikan berdasarkan aksioma yang sudah ada dan berdasarkan pada premis dan kesimpulan. Peterson menjelaskan bahwa ahli matematika Jerman besar David Hilbert menganjurkan program ambisius untuk merumuskan sistem aksioma dan aturan kesimpulan yang akan mencakup semua matematika, dari aritmatika dasar untuk kalkulus lanjutan. Hempel, CG, 2001, mencatat bahwa, dalam matematika, teorema teori apapun sebagai suatu peraturan dari dua bagian yaitu premis dan kesimpulan. Oleh karena itu, kesimpulan dari teorema berasal tidak hanya dari satu set tetap aksioma, tetapi juga dari premis yang khusus untuk teorema tertentu; dan premis ini tidak merupakan perpanjangan dari sistem tetap prinsip.