The choice to be made is similar to the choice between set theory and category theory.
@Tadeusz Ostrowski: In her [Mathematics] palace there are rooms for all theories.
In addition to each of the incisive (and interesting) answers, already given, the answer given by @Tadeusz Ostrowski (and @George Stoica) makes perfect sense: rather than choosing between set theory and topos theory as a foundation of mathematics, we can view Mathematics as a form of Hilbert Hotel (there is always room for one more guest). In terms of this thread, a new guest would be a fundamentally important theory in the foundations of mathematics.
I myself favour axiomatic set theory, especially if we start considering various forms of topology.
Category theory used topoi as a way to approach sets, alternative to set theory. This way, category theory found an alternative foundation for mathematics. Set theory is a foundation for mathematics too and was the focus of the other question.
To my mind the answer depends on intentions.
Set theory demonstrated its efficiency for the part of mathematics that serves to justify physical theories. But this approach is not efficient to justify information science: the last one uses non-classical and modal logics which do not have set-theoretic models. And topoi are appropriate in the context.
The complicated question is can we obtain meaningful theories based on this kind of generalisations?
I would choose model-theory - in most situations first order, but if necessary also second-order. Of course I have a small foundational circularity, because in order to make model-theory I need a notion of set, but this problem is in fact unavoidable, whatever I choose. On the other hand model-theory helps me to study also models of set-theory or models of category theory, as like models of logic or models of model-theory itself. I like to figure in my mind every page cut in two columns, the left one containing semantic and formal statements, the right one classes of models and morphisms or other relations between them. This is the way I like to think about foundation of doesn't matter what. Nothing is purely substantial (semantic) or purely formal (sintactic). Mathematics is a permanent back and forth between those two aspects.
I believe for reasons of mathematic severity that better is the topoi theory.
Yiannis Theodorou
Prof. of Math.- Technological University of central Greece
What is your goal(s)? Pedagogical (in maths) but then at what level? Philosophical? ...
César Rodrigues wrote :What would you choose as a foundation for mathematics, set theory or topos theory.?
In their existing form nothing from both.
It would be wonderful to find an answer to the complicated question presented by Grygoriy.
I think we do not have to choose between set theory and topos theory. We have different geometries, logics, set theories (with axiom choice and without it), and so on. Thanks to that mathematics is such a rich, powerful and interesting queen of science. In her palace there are rooms for all theories.
Greetings, Tadeusz
Regarding foundations of mathematics, Type Theory seems to me the best approach. Category theory and Set Theory can be constructed from it. Several variants of the theories exist. I found the Homotopy Type Theory is the most advanced. It does not rely on the law of excluded middle.
The main advantage of Type Theory is its stronger link (not dependency) to Computer Science: the most spectacular achievements are the proof assistants (such Coq) and certified programming (such the CompCert-C compiler and several micro-kernels). Such frameworks are treasures for applied mathematics.
http://homotopytypetheory.org/
https://coq.inria.fr/
http://www.cis.upenn.edu/~bcpierce/sf/current/
https://ncatlab.org
http://compcert.inria.fr/
@Tadeusz Ostrowski: In her [Mathematics] palace there are rooms for all theories.
In addition to each of the incisive (and interesting) answers, already given, the answer given by @Tadeusz Ostrowski (and @George Stoica) makes perfect sense: rather than choosing between set theory and topos theory as a foundation of mathematics, we can view Mathematics as a form of Hilbert Hotel (there is always room for one more guest). In terms of this thread, a new guest would be a fundamentally important theory in the foundations of mathematics.
I myself favour axiomatic set theory, especially if we start considering various forms of topology.
>I myself favour axiomatic set theory, especially if we start considering various forms of topology.
In the usual mathematics we use something more than Con(ZFC), but
Con[ZFC+(E)(omega model of ZFC)].Thus it is constructed on sand
There is a standard opinion that any changing of the foundations are necessary only after collapse
>it is a question in philosophy-of-mathematics
No at all. For example concert of topoi are necessary for the constructive non-standard analysis
Set Theory as it has a clear and convincing axiomatization which is not the case in the category theory.
Note that questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical constructions are permissible.
Shouldn't you choose Philosophy and the Ancient Philosophers as the basis for all Mathematics instead?
These both foundation uses ancient representation about countable infinity such that:there is a set which intuitively has infinitely many elements
0,{0},{{0}},...etc.
And it is strange.
In additional both foundations are based on abnormal belief that infinite sequence such that 0,{0},{{0}},...etc gives a model of Peano postulates.
@ Marcia Ricci Pinheiro
In fact contemporary foundations based on ancient philosophy of infinity and ancient logic.
With set theory consistent and category consistent in the context of a plural mathmatics we are in a best world, that was lacking since the foundational crises of the begin of last century.
If ZFC is inconsistent, but the inconsistency is not very finite, then it
doesn't matter, because we will never discover it. If instead ZFC proves
"there is a proof of inconsistency of ZFC of length less than 10^4000" then
I'm not sure what I'd make of it.
Both aspects of Math theory mentioned in the question are important. However, in order to know what set a topological structure is defined on, we should define firstly that set. So, in my opinion, the natural order is: basic notions on set theory; basics of topology; compatible algebraic -topological structures (such as topological groups, topological vector spaces, etc.). Another natural and important notion, with deep applications, is that of an order relation. Such a relation is useful when it is compatible with some other structures. Most of the spaces in analysis have a natural topological ordered vector space structure. For example C(K) (the space of real continuous functions on a compact topological space K) is a Banach lattice and a Banach algebra too. The bicommutant is an order complete Banach lattice and a commutative Banach algebra of selfadjoint operators acting on a Hilbert space. Some more information and examples can be found in "Topological Vector Spaces", by H. H. Schaefer. See also "Ordered Vector Spaces and Linear Operators" by Romulus Cristescu,, Academiei, Bucharest, Romania and Abacus Press, Tunbridge Wells, Kent, England.
One of the diferences of set theory and category theory is that set theory began in 19th century and category theory began a century later, the 20th century. A big difference is that set theory is subject to the incompleteness theorems of Goedel and category theory seems to not being affected. But Godel theorems were subject to a process of relativization, first the first Godel theorem then the second Godel Theorem by myself. See questiona "is the set theory (ZFC) consistent?", in my profile. We can procede not worrying with Godel theorems.
The search for an overarching foundation for mathematics doesn´t seem a good idea, neither for philosophy nor for mathematics, specially if such a foundation is one among many other branches that make up mathematics. Mathematics changes over time, in complexity and richness, in its discipline boundaries, so perhaps it would be more rewarding, both for philosophy and for mathematics, to understand and further this development. An all-embracing foundation guided by narrow philosophical concerns doesn´t seem a good idea either--this seems to happen sometimes when philosophical quarrels among "isms" (logicisms, platononi, formalism, structuralism, etc.) underscore philosophical discourse about mathematics.
Philosophy has interesting means for reflecting upon itself and thereby changing its self-image, its lines of enquiry and so forth.
Something similar seems to happen in mathematics sometimes, when old concepts, problems and theories are recasted in novel and interesting ways by means of unforeseen tools provided by new theories, problems, solutions, etc.
Another thing that would perhaps be rewarding is to broaden the notion of mathematical theory. It seems that it is isually taken in the sense of being merely given by sequences like axioms-definitions-theorems, when in practice things fortunately do not seem to be that simple.
@ César Rodrigues
I do not know for what mathematics so strongly insist on consistency.The theory of inconsistent sets is offered by logicians not yesterday.
Forgive me, even Google arrived to a positive answer respecting the consistency of set theory. Go to my answer on this topic.
We may speak of set theory realism but it is more difficult to refer category theory realism.
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