Oct 10, 2012

Elegi Menggapai "Mathematical Truth"




By Marsigit
Yogyakarta State University


It 1 is usually thought that mathematical truths are necessary truths. Two main views are possible i.e. either they are known by reason or they are known by inference from sensory experience.



The former rationalist view is adopted by Descartes and Leibniz who also thought that mathematical concepts are innate; while Locke and Hume agreed that mathematical truths are known by reason but they thought all mathematical concepts were derived by abstraction from experience.

Mill 2 was a complete empiricist about mathematics and held that mathematical concepts are derived from experience and also that mathematical truths are really inductive generalizations from experience.

Weir A. theorizes that one obvious problem for neo-formalism is its apparent conflict with Gödel's first incompleteness result showing that not all mathematical truths are provable, under a certain conception of provability.

Even though the neo-formalist 3 makes no synonymy claim between 'sixty eight and fifty seven equals one hundred and twenty five' and "68+57=125" is provable', this result seems to rule out any tight equivalence between truth and proof of the sort envisaged.

According to Weir A., we need a distinction between legitimate and illicit transformations, if neo-formalism is to avoid the consequence that in mathematics there is no distinction between truth and falsity.

It 4 cannot be that a string is provable if derivable in the one true logic from some consistent set of axioms or other; even if there is only one true logic it would still follow that any logically consistent sentence i.e. a mathematical truth.

The neo-formalist, as it was notified by Weir A., perceives that provability in a practice means derivable using only inference rules which are in some sense analytic and constitutive of the meaning of our logical and mathematical operators.

There are the responses to the neo-formalist, that no rule can be meaning-constitutive if it is trivial. The inconsistency and indeed triviality of 'classical' naïve set theory 5 is a product of three things: the classical operational rules, the classical structural rules and the naïve rules or axioms.

The neo-formalist 6 agrees with the strict finitist that the only objects with a title to being called mathematical which exist in reality are the presumably finite number of concrete mathematical utterances; some of these utterances, however, are used to assert that infinitely many objects- numbers, sets, strings of expressions, abstract proofs, etc.- exist.

Mathematical truth 7 is thus linked with provability in formal calculi and in such a way as to be perfectly compatible with the claim that all that exists in mind-dependent reality are concrete objects together with their physical properties.

Field H. observes that the determinacy of mathematical statements is primarily dependent on the precision we can give to the semantics of the language in which they are expressed.

If 8 we are dealing with mathematics expressed in first order logic, then the semantics of the logic itself are pretty well nailed down and if the theory under consideration concerns a unique structure up to isomorphism then we know that each closed sentence will have a definite truth value under that interpretation, and there will only be indeterminacy if there is some substantive ambiguity about what this unique intended interpretation is.


Field H. 9 , claims that even if as in the case of arithmetic, there can be no complete recursive axiomatization of the theory, which will normally be the case where there is a unique intended interpretation.

On the other hand, Oddie G. says:

While mathematical truth is the aim of inquiry, some falsehoods seem to realize this aim better than others; some truths better realize the aim than other truths and perhaps even some falsehoods realize the aim better than some truths do. The dichotomy of the class of propositions into truths and falsehoods should thus be supplemented with a more fine-grained ordering -- one which classifies propositions according to their closeness to the truth, their degree of truth-likeness or verisimilitude. The problem of truth-likeness is to give an adequate account of the concept and to explore its logical properties and its applications to epistemology and methodology. 10

Popper 11 refers to Hume’s notion that we not only that we can not verify an interesting theory, we can not even render it more probable.

There 12 is an asymmetry between verification and falsification and while no finite amount of data can verify or probability an interesting scientific theory, they can falsify the theory.

Popper 13 indicates that it is the falsifiability of a theory which makes it scientific; and it implied that the only kind of progress an inquiry can make consists in falsification of theories.

Popper states that if some false hypotheses are closer to the truth than others, if verisimilitude admits of degrees, then the history of inquiry may turn out to be one of steady progress towards the goal of truth.

It may be reasonable, on the basis of the evidence, to conjecture that our theories are indeed making such progress even though it would be unreasonable to conjecture that they are true simpliciter. 14

Again, Oddie G. convicts that the quest for theories with high probability must be quite wrong-headed, while we want inquiry to yield true propositions, in which not any old truths will do.

A tautology 15 is a truth, and as certain as anything can be, but it is never the answer to any interesting inquiry outside mathematics and logic. What we want are deep truths, truths which capture more rather than less, of the whole truth.

Even more important, there is a difference between being true and being the truth. The truth, of course, has the property of being true, but not every proposition that is true is the truth in the sense required by the aim of inquiry.

The truth of a matter at which an inquiry aims has to be the complete, true answer.

Oddie G. illustrates the following:

The world induces a partition of sentences of L into those that are true and those that are false. The set of all true sentences is thus a complete true account of the world, as far as that investigation goes and it is aptly called the Truth, T. T is the target of the investigation couched in L and it is the theory that we are seeking, and, if truthlikeness is to make sense, theories other than T, even false theories, come more or less close to capturing T. T, the Truth, is a theory only in the technical Tarskian sense, not in the ordinary everyday sense of that term. It is a set of sentences closed under the consequence relation: a consequence of some sentences in the set is also a sentence in the set. T may not be finitely axiomatisable, or even axiomatisable at all. The language involves elementary arithmetic it follows that T won't be axiomatisable; however, it is a perfectly good set of sentences all the same. 16

If a mathematical truth 17 is taken to be represented by a unique model, or complete possible world, then we end up with results very close to Popper's truth content account.

In particular, false propositions are closer to the truth the stronger they are; however, if we take the structuralist approach then we will take the relevant states of affairs to be “small” states of affairs viz. chunks of the world rather than the entire world and then the possibility of more fine-grained distinctions between theories opens up.

Further, Oddie G. (2001) claims that a theory can be false in very many different ways; the degree of verisimilitude of a true theory may also vary according to where the truth lies.

One does not necessarily make a step toward the truth by reducing the content of a false proposition.

Nor does one necessarily make a step toward the truth by increasing the content of a false theory.

References:
Philosophy of Mathematics, Retrieved 2004
2 Ibid
3 Weir A., 2004, “A Neo-Formalist Approach to Mathematical Truth”, Retrieved 2004
4 Ibid.
5 Ibid.
6 Ibid.
7 Ibid.
8 Field, H., 1999, “Which Undecidable Mathematical Sentences Have Determinate Truth Values?”, RJB, Retrieved 2004
9 Ibid.
10 Oddie, G., 2001, “Truthlikeness”, Stanford Encyclopedia of Philosophy, Paideia, Philosophy of Mathematics. Retrieved 2004
11Popper in Oddie, G., 2001, “Truthlikeness”, Stanford Encyclopedia of Philosophy, Paideia, Philosophy of Mathematics. Retrieved 2004
12Oddie, G., 2001, “Truthlikeness”,, Stanford Encyclopedia of Philosophy, Paideia, Philosophy of Mathematics. Retrieved 2004
13Popper in Oddie, G., 2001, “Truthlikeness”, Stanford Encyclopedia of Philosophy, Paideia, Philosophy of Mathematics. Retrieved 2004
14Ibid.
15Oddie, G., 2001, “Truthlikeness”, Stanford Encyclopedia of Philosophy, Paideia, Philosophy of Mathematics. Retrieved 2004
16Ibid.
17Ibid.

12 comments:

  1. Heni Lilia Dewi
    16709251054
    Pascasarjana Pendidikan Matematika 2016/ Kelas C

    I pointed that mathematical truths categorize in two major line, they are known by reason or they are known by inference from sensory experience. It is undeniable that there are varoius perceptions about mathematical truths. As we know, some agreed that mathematical concepts are innate; while others agreed that mathematical truths are known by reason but they thought all mathematical concepts were derived by abstraction from experience. Mathematical truths are also about either known by reason or known by inference from sensory experience.

    ReplyDelete
  2. Cendekia Ad Dien
    16709251044
    PPs Pendidikan Matematika Kelas C 2016

    Kebenaran matematika dilihat dari dua sudut pandang utama yang dikenal dengan akal atau inferensi yang berasal dari pengalaman indrawi. Kaum rasionalis menganggap konsep matematika itu berasal dari akal namun semua konsep matematika diperoleh abstraksi yang berasal dari pengalaman. Adapun kaum empiris menganggap konsep matematika itu berasal dari pengalaman sehingga kebenaran matematika benar-benar merupakan generalisasi induktif dari pengalaman. Sementara neo-formalis menganggap kebenaran matematika dapat dibuktikan melalui logika.

    ReplyDelete
  3. Wahyu Lestari
    16709251074
    PPs Pendidikan Matematika 2016 Kelas D

    dari artikel di atas, kebenaran matematika adalah tujuan penyelidikan, beberapa kepalsuan tampaknya menyadari tujuan ini lebih baik daripada yang lain; Beberapa kebenaran lebih baik mewujudkan tujuan daripada kebenaran lain dan bahkan mungkin beberapa kesalahan membuat tujuan lebih baik daripada beberapa kebenaran. Dikotomi kelas proposisi menjadi kebenaran dan kepalsuan karenanya harus dilengkapi dengan urutan yang lebih halus - yang mengklasifikasikan proposisi sesuai dengan kedekatan mereka dengan kebenaran, tingkat kebenaran atau kemiripannya. Masalah kemiripan kebenaran adalah dengan memberikan konsep yang memadai tentang konsep tersebut dan untuk mengeksplorasi sifat logis dan aplikasinya terhadap epistemologi dan metodologi.

    ReplyDelete
  4. Sumandri
    16709251072
    S2 pendidikan Matematika 2016

    Kebenaran matematika dasar tergantung pada pembenaran pembuktian matematis. Hal ini pada gilirannya tergantung pada asumsi sejumlah pernyataan matematika dasar (aksioma), serta pada logika yang mendasarinya. Secara umum, pengetahuan matematika terdiri dari pernyataan yang dibenarkan oleh bukti, yang tergantung pada aksioma matematika (dan logika yang mendasarinya). Logika yang mendasari tidak ditentukan selain pernyataan dari beberapa aksioma tentang hubungan kesetaraan.. Aksioma yang tidak dianggap sebagai asumsi sementara diadopsi, yang digunakan hanya teori berdasarkan pertimbangan. Aksioma yang menjadi dasar kebenaran tidak diperlukan adanya pembenaran. Karena itu, bukti logis mempertahankan kebenaran dan diasumsikan aksioma adalah kebenaran yang jelas, maka setiap teorema yang berasal darinya juga harus benar .

    ReplyDelete
  5. PUTRI RAHAYU S
    S2 PENDIDIKAN MATEMATIKA_D 2016
    16709251070

    Secara epistemologis kebenaran adalah kesesuaian antara apa yang diklaim sebagai diketahui dengan kenyataan yang sebenarnya yang menjadi objek pengetahuan. Kebenaran matematika membutuhkan pembuktian dan kesepakatan. Ada dua teori tentang kebenaran dalam matematika, yaitu teori korespondensi dan teori koherensi. Teori kebenaran korespondensi adalah teori yang berpandangan bahwa pernyataan-pernyataan adalah benar jika berkorespondensi terhadap fakta atau pernyataan yang ada di alam atau objek yang dituju pernyataan tersebut. Kemudian teori kebenaran koherensi berpandangan bahwa suatu pernyataan dikatakan benar bila terdapat kesesuaian antara pernyatan satu dengan pernyataan terdahulu atau lainnya dalam suatu sistem pengetahuan yang dianggap benar.

    ReplyDelete
  6. Sehar Trihatun
    16709251043
    S2 Pend. Mat Kelas C – 2016

    Kebenaran tentang matematika selalu menjadi bahasan yang menarik untuk dibahas. Para filsufpun tak luput dari pembicaraan seputar kebenaran matematika. Setidaknya ada dua pandangan besar yang terkait dengan kebenaran matematika yang dikemukakan oleh beberapa filsuf ini yakni yang pertama, bahwa kebenaran matematika dapat diketahui melalui penalaran dan yang kedua, kebenaran matematika dapat diketahui melalui inferensi langsung dengan pengalaman indrawinya.

    ReplyDelete
  7. Anwar Rifa’i
    PMAT C 2016 PPS
    16709251061

    Oddie G mengatakan " Ketika kebenaran matematika bertujuan inquiry, beberapa yang salah menyadari bahwa tujuan ini lebih baik dibanding yang lain, beberapa kebenaran lebih baik menyadari tujuan dibanding kebenaran lain dan mungkin beberapa yang salah menyadari tujuannya lebih baik dibanding yang lain. Dikotomi dari kelas Preposisi kepada kebenaran dan salah harus ditambahkan dengan urutan penarikan kesimpulan - seseorang yang mengklasifikasikan proposisi berdasarkan kedekatannya pada kebenaran, tingkat kebenarannya atau seakan-akan terlihat benar.

    ReplyDelete
  8. Lihar Raudina Izzati
    16709251046
    P. Mat C 2016 PPs UNY

    Ada dua teori tentang kebenaran dalam matematika, yaitu teori korespondensi dan teori koherensi. Teori kebenaran korespondensi berpandangan bahwa pernyataan-pernyataan adalah benar jika berkorespondensi terhadap fakta atau pernyataan yang ada di alam atau objek yang dituju pernyataan tersebut. Teori kebenaran koherensi berpandangan bahwa suatu pernyataan dikatakan benar bila terdapat kesesuaian antara pernyatan satu dengan pernyataan terdahulu atau lainnya dalam suatu sistem pengetahuan yang dianggap benar.

    ReplyDelete
  9. Sylviyani Hardiarti
    16709251069
    S2 Pendidikan Matematika Kelas D 2016

    Kebenaran matematika membutuhkan pembuktian dan kesepakatan. Kebenaran matematika dapat berasal dari akal atau kesimpulan dari pengalaman indrawi. Kebenaran matematika berasal dari konsep-konsep matematika yang diperoleh melalui abstraksi dari pengalaman. Selain itu, matematika bukanlah suatu pengetahuan yang tidak memerlukan pembuktian lagi. Adanya konjektur matematika menunjukkan bahwa tidak semua kebenaran matematika sudah jelas atau pasti nilai kebenarannya. Konjektur adalah sebuah pernyataan matematika yang mana nilai kebenarannya tidak atau belum kita ketahui. Untuk membuktikan konjektur tersebut, maka pastinya dikaitkan dengan dalil-dalil dasar matematika dan proses pembuktian tersebut tidaklah gampang serta mengalami proses yang panjang.

    ReplyDelete
  10. Primaningtyas Nur Arifah
    16709251042
    Pend. Matematika S2 kelas C 2016
    Assalamu’alaikum. Kebenaran matematika adalah kebenaran menurut definisi atau persyaratan yang menentukan makna dari istilah kunci. Kebenaran matematika semata-mata dapat ditunjukkan dengan menganalisis makna yang terkandung dalam istilah di dalamnya, yang di dalam logika disebut sebagai benar secara apriori yang mengindikasikan bahwa nilai kebenarannya bebas secara logis dari atau apriori secara logis pada sebarang bukti eksperimental.

    ReplyDelete
  11. Desy Dwi Frimadani
    16709251050
    PPs Pendidikan Matematika Kelas C 2016

    Kebenaran matematika adalah kebenaran yang membutuhkan bukti dan kesepakatan, dengan pembuktian dan kesepakatan secara matematis. Dalam kebenaran matematika terdapat dua teori kebenran yaitu teori korespondensi dan teori koherensi

    ReplyDelete
  12. Helva Elentriana
    16709251068
    PPS Pend Matematika Kelas D 2016

    Ada dua pandangan yang membahas tentang kebenaran matematika. Pandangan rasionalis sebelumnya diadopsi oleh Descartes dan Leibniz yang juga berpikir bahwa konsep matematika itu bawaan; Sementara Locke dan Hume sepakat bahwa kebenaran matematika dikenal dengan akal tetapi mereka pikir semua konsep matematika diturunkan oleh abstraksi dari pengalaman.

    ReplyDelete