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. Nur Dwi Laili K
    PPs Pendidikan Matematika C

    Sesungguhnya selama saya mempelajari matematika, nama Hempel dan Peterson merupakan nama yang baru bagi saya. Hal ini tentu kembali menampar saya bahwa ilmu yang saya pelajari selama ini hanyalah sebagian kecil dari ilmu yang ada.
    Pada tulisan Prof. Marsigit disebutkan Hempel, C.G., 2001, memberitahukan bahwa sistem dasar prinsip mandiri yang stabil adalah ciri khas dari teori matematika. Hal ini mengingatkan saya bahwa matematika murni merupakan ilmu langit dimana segala definisi, teorema, dan aksioma yang ada adalah benar asalkan tidak bertentangan atau tidak terdapat kontradiksi. Tetapi ini jelas berbeda dengan matematika sekolah dimana matematika sekolah dipelajari melalui pengalaman yang artinya tidaklah stabil karena pengalaman antara satu anak dengan lainnya pastilah berbeda.

  2. This comment has been removed by the author.

  3. Nama: Hendrawansyah
    NIM: 17701251030
    S2 PEP 2017 Kelas B

    Assalamualaikum wr wb

    Jikalau tidak salah memahami, saya menangkap bahwa Peterson dan Hempel berusaha menjadikan aksioma sebagai raja di dalam pembuktian terkait pernyataan di dalam matematika.Aksioma ingin dijadikan motif tunggal dalam penalaran matematika.Kedua ilmuwan tersebut menawarkan konsep murni bahwasannya kebenaran matematika hanya dapat diturunkan melaui aksioma. Hal ini tentu mengalami pertentangan dengan konsep idealitas.Konsep idealitas dipandang perlu untuk mengetahui apa yang melatarbelakangi asumsi yang demikian.Dan logika memiliki peran untuk merepresentasikan hal tersebut.

  4. Nama: Dian Andarwati
    NIM: 17709251063
    Kelas: Pendidikan Matematika (S2) Kelas C

    Assalamu’alaikum.Hempel menegaskan bahwa matematika merupakan ilmu empiris yang berbeda dari cabang lain seperti astronomi, fisika, kimia, dll, terutama dalam dua hal: materi pelajaran adalah lebih umum daripada apapun lainnya dari penelitian ilmiah, dan proposisi yang telah diuji dan dikonfirmasi ke tingkat yang lebih besar dibandingkan beberapa bagian yang paling mapan astronomi atau fisika. Dengan demikian, sejauh mana hukum-hukum matematika telah dibuktikan oleh pengalaman masa lalu umat manusia begitu luar biasa bahwa kita telah dibenarkan olh teorema matematika dalam bentuk kualitatif berbeda dari hipotesis baik dari cabang lain.

  5. Yusrina Wardani
    PPs PMAT C 2017
    Matematika berasal dengan bukti matematis dari alam yang terbatas.
    Hal ini berarti bukti dapat diperoleh dari pengalaman manusia yang berinteraksi dengan lingkungan tapi aman kebenaran. Dalam belajar matematika, inovasi yang ada harus disesuaikan dengan karakteristik semua yang terlibat dalam proses pembelajaran. Berikutnya hasil pemikiran tersebut diwujudkan atau diimplementasikan yang secara berkelanjutan perlu dimonitoring dan dievaluasi sebagai sebuah sistem.

  6. Tri Wulaningrum
    PEP S2 B

    Berdasarkan artikel di atas, saya melihat bahwa menggapai kebenaran pengetahuan dalam matematika terdapat tarik ulur terhadap hukum identitas. Hal itu sesuai dengan pernyataan Hilbert bahwa sistem aksioma dan peraturan formal harus konsisten, yang berarti bahwa seseorang tidak dapat membuktikan sebuah pernyataan dan kebalikannya pada saat bersamaan. Tarik ulru terhadap hukum identitas tentunya juga akan melahirkan hukum kontradiktif. Sesuai dengan pernyataan Hilbert tersebut bahwa tidak dapat dibuktikan suatu pernyataan dan sebalikanya dalam waktu yang bersamaan. Hal itu dikarenakan ketika kita berhasil membuktikan suatu pernytaan, maka sudah pasti kebalikan dari pernyataan tersebut tidaklah terbukti.

  7. Junianto
    PM C

    Tujuan memepelajari matematika mulai dari aritmetika dasar hingga kalkulus lanjut adalah bagaimana agar bisa disusun metode penalaran matematis dan memasukkannya dalam kerangka tunggal. Mengutip pendapat Hilbert2 yang mengatakan bahwa harus ada prosedur mekanis yang jelas untuk menentukan apakah suatu proposisi tertentu mengikuti serangkaian aksioma tertentu. Sehingga, ketika ada prosedur teknis maka harapannya pembuktian yang dilakukan akan lebih kredibel dan bisa diterima secara umum.

  8. Latifah Fitriasari
    PM C

    Matematika dapat dijadikan sebagai acuan dasar untuk mempelajari matematika yang lebih dalam. Matematika merupakan ilmu tentang kuantitas, struktur, ruang, dan perubahan. Matematikawan menemukan pola, merumuskan dugaan baru, dan membangun kebenaran melalui metode deduksi ketat yang berasal dari aksioma dan definisi bertepatan. Menurut Hempel validitas matematika berasal dari ketentuan yang menentukan arti dari konsep-konsep matematika,dan bahwa proposisi matematika karena itu pada dasarnya benar menurut definisi.

  9. Anisa Safitri
    PEP B

    Petersons mengatakan bahwa matematika itu lemah untuk menjawab pertanyaan pertanyaan tertentu. Pengambilan kesimpulan yang berlaku dalam konsep matematika tidaklah jauh dari logika. Logika yang berkembang juga meliputi pemikiran rasional. Matematika itu ilmu yang struktur. dari Hampel matematika itu diterima dengan valid sesuai dengan indiktor-indikaor dalam matematika itu sendiri. seperti darikonsep matemtika, definisi yang harus dibuktikan.

  10. Isoka Amanah Kurnia
    PPs Pendidikan Matematika 2017 Kelas C

    According to Hempel based on the principles and rules of formal logic, the contents of all mathematics can be derived from simple systems that are remarkable achievements in mathematical content systems and clarify the basis of validity. In everyday life as well as in scientific language, Hempel insists that mathematics is to be a correct theory of the concepts in question.

  11. Auliaul Fitrah Samsuddin
    PPs P.Mat A 2017
    Terima kasih atas postingannya Prof. Seperti yang diketahui bahwa menurut David Hilbert matematika adalah sistem logika yang konsisten dan formal. Peterson menyatakan bahwa yang dikatakan Hilbert maksudnya adalah kita dapat menyelesaikan madalah jika kita cukup cerdas dan mengerjakannya dalam waktu yang lama; menurut Chaitin dan Watson percaya bahwa tidak ada yang tidak dapat dicapai oleh Matematika.

  12. Nama : Habibullah
    NIM : 17709251030
    Kelas : PM B (S2)

    Assalamualaikum wr.wb

    Matematika sebagai bahasa dan pengetahuan universal menjadi dasar bagi perkembangan teknologi modern. Selain itu, matematika juga merupakan pengetahuan yang dapat mengakomodir kemampuan berfikir, sebab cirri khas dari matematika yaitu proses berlanar dan penarikan kesimpulan yang logis, sehingga tidak hanya bermanfaat pada pengetahuan matematika itu sendiri melainkan juga pada kehidupan sehari-hari. Melihat pentingnya matematika, maka guru memiliki peranan dalam memfasilitasi siswa dalam proses berfikir secara logis, analitis, sistematis, kritis dan kreatif.

  13. Atik Rodiawati
    S2 Pendidikan Matematika B 2017

    Bacaan inilah yang membuktikan pada kita bahwa matematika memang dikembangkan secara analitis, dan inilah sifat matematika. Matematika berawal dari definisi dan semua teorema dan aksioma yang berkembang di bawahnya harus sesuai atau tidak bertentangan dengan definisi awal tersebut atau disebut sifat matematika itu konsisten. Bacaan ini juga memberikan gambaran pada saya bahwa dalam kesepakatan pun matematika berkembang dinamis, dimana banyak ilmuwan yang saling bertentangan, namun hal ini terjadi karena sesuai tujuan matematika itu sendiri yakni mengembangkan penalaran dan pembuktian matematis para matematikawan.