THE FOUNDATIONS OF MATHEMATICS
Arranged by:
LENNY PUSPITA DEWI
05301244122
MATHEMATICS EDUCATION
FACULTY OF MATHEMATICS AND NATURAL SCIENCE
YOGYAKARTA STATE UNIVERSITY
2008
LENNY PUSPITA DEWI
05301244122
MATHEMATICS EDUCATION
FACULTY OF MATHEMATICS AND NATURAL SCIENCE
YOGYAKARTA STATE UNIVERSITY
2008
Mathematics is a logical science, cleanly structured, and well-founded. Here we look at those foundations.
A. What is a "foundation" for mathematics?
We discuss some of the ways the word "foundation" is used in relation to mathematics.
· Branch Foundations
Each branch or field of mathematics may have its own foundational elements, special to the field. These may be the fundamental concepts investigated in the field, fundamental results on which most other results are based, or pervasive methods.
· Fundamental Concepts
There are some mathematical concepts which pervade not just one branch of mathematics but the whole of mathematics. An obvious example is the concept of a function or that of a homomorphism. Variation in these concepts may either provide different ways of doing mathematics (e.g. category theory), or may lead to different kinds of mathematics (e.g. constructive mathematics).
B. Logical Foundations for Mathematics
The methods of mathematics are deductive, and logic therefore has a fundamental role in the development of mathematics. Suitable logical frameworks in which mathematics can be conducted can therefore be called logical foundation systems for mathematics.
· Paradigms
There are many alternative logical foundations for mathematics. They differ in some cases just in detail or strength. In other cases the differences are more fundamental, possibly representing radically divergent views on the nature of mathematics.
· Formal
The formal aspect concerns the mathematical theorems which can be proven using the foundation system. Foundation systems are partially ordered according to their proof-theoretic strength.
· Logical
In some cases it is possible to separate out a part of the system which is concerned with logic and independent of matters ontological.
· Ontological
Ontology is an important part of semantics, and differences both paradigmatic and in detail, is likely to be reflected in or caused by ontology.
· Dimensions
We consider five different characteristics or dimensions of logical foundation systems, sometimes clearly separated, other times not so. The formal the semantic, the logical, the ontological and the conceptual.
· Semantic
Gödel's first incompleteness theorem guarantees that truth and theorem hood do not coincide in any foundation system adequate for mathematics. It is therefore desirable to have an account of the intended meaning of the language independent of the definition of formal derivability.
· Conceptual
Over and above the logical and ontological features which determine the strength of the system there is likely to be some conceptual apparatus which provides the first stages in developing mathematics.
3 comments:
Good Lenny, anda telah dengan cepat merespon saya. Saya pikir lebih baik begitu, anda menunjukkan foto asli anda. Tetapi saya masih merisaukan judul blog anda dengan istilah "Mad". Itu menunjukkan anda tidak suka matematika. Apa memang demikian? Coba ditata kembali pikiran, perasaan dan kemampuan. Cobalah dari yang kecil saja tetapi bersifat positif. Tulisan anda yang lalu yang dalam bahasa Inggris juga agak membingungkan karena pembaca akan ragu apakah itu pendapat anda atau orang lain. Jika masih sulit dalam Bahasa Inggris coba ungkapkan saja dalam Bahasa Indonesia agar orang lain mengetahui pikiran dan ide orisinal anda. Selamat berjuang. Amien (Dosen: Dr. Marsigit)
Pendapat anda yang humble tentang Goggle tidak begitu jelas. Dan kira-kira penting bagi saya. Coba anda sampaikan saja dalam bahasa Indonesia. Kalau tidak salah anda menyarankan agar saya tidak konek ke ... sehingga identitas anda tidak otomatis muncul. Apa begitu. Tolong kalau bisa diulangi lagi saran anda agar bisa saya tindaklanjuti. Trim (Dosen: Dr. Marsigit)
Hei,,,
I think its good post by explaining 'bout the foundation of mathematics.
So, can u tell me how significant important for us to know the foundation of mathematics?
And can u make sure that it is applicable in our study?
Thanks for the respond....^^
Ow ya....u're mad of math...so do I...but I'm mad of love it....^^
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