We observed in §1.6 that category theory is adept at capturing the significant ideas in a mathematical discipline in the form of universal properties, whilst axioms may be expressed in symbolic logic as systems of introduction and elimination rules. Then in §2 we demonstrated the formal connection between these.

**3.1.**
On this basis, we propose a methodology for devising a new
foundational system *for* a subject.
This is summed up by the diagram

In this, the task of the journeyman mathematician — the deduction of theorems from axioms — is to fill in the upward arrow on the right. Hopefully, this will make the rectangle “commute” in the sense that we recover the old theorems from the new axioms.

However, this job only makes sense in a professional framework
in which axioms have *already* been chosen to capture the intuitions
and applications of the discipline.
This choice is represented by the downward arrow on the left.

The constructions that we sketched in §§2.8f provide the horizontal arrow in the middle, whilst the computational interpretation (bottom right) depends on the details of the resulting calculus (§2.3).

We shall show how this programme works for the example of general topology in the remaining sections of this paper , with some hints about how to apply the ideas to other subjects.

**3.2.**
The first arrow in some sense derives the axioms from the theorems.

Euclid and Bourbaki gave us a style of presenting mathematics
that begins from axioms which (so far as their text is concerned)
come out of the blue,
and deduces theorems from them as if the results were obvious and inevitable.
Abel described Gauss as “like a fox, who effaces his tracks
in the sand with his tail”.
Consequently, lesser mathematicians think that
it is enough to *state* a bunch of axioms in order
to justify the *relevance* of their results,
whilst students learn nothing about the ways in which their masters
*discovered* their theorems.

In following our methodology, we cannot “write down all the axioms
first”, because *finding* the right axioms is the
*goal* of the investigation.
Being able to deduce the old theorems from the new axioms is an
*experimental test* of the axioms that we propose.
So before we have discovered the *final version* of the axiom system,
we need to conduct preliminary experiments in *ad hoc* frameworks
(*cf*. §5).

Along the way, like any engineer, we shall assemble several *prototypes*.
This means that a certain amount of *repetition* of tests in needed.
Of course this happens all the time in mathematics too [Lak63] —
it is just the textbooks that falsify history by saying otherwise.
At the end of the day,
we want to state our chosen axioms and deduce the important theorems from them.
But that will indeed be the *end* of the process,
which will then no longer be an active piece of *foundational* research.

For a modern study of a wide range of techniques for discovering the principles of several mathematical disciplines, see [Cor03].

**3.3.** Counterexamples play a major part in the empirical
development of mathematics. Unfortunately, they are sometimes given such
a degree of prominence that they can stifle subsequent progress.

Typically, the argument behind a counterexample may prove

D, E, F ⊢ ¬ G, |

entirely rigorously. Then the reader is expected to agree that,

therefore, E is false, |

but why? *D*, *F* and *G* had exactly the same status in the proof,
so why pick on *E*?

This is an argument *by authority*, not a valid piece of mathematics:
the author has simply used sleight-of-hand to make us accept his prejudices.
These become embedded in the literature
and treated as if they were *theorems*,
on which whole theories are built.
Having forgotten that the counterexample depended on cultural assumptions,
people subsequently resist the introduction of a new paradigm.

This abuse of counterexamples is very common,
but the one that has caused the greatest intellectual paralysis
is surely Georg Cantor’s, that no set is in bijection with its powerset.
On this he built his theory of cardinality,
which is far too coarse to be of any use in mainstream mathematics.
He said of the discovery that ℝ^{2}≅ℝ,
“I see it but I don’t believe it”,
but this did not deter him from jumping into the abyss.

Because of Cantor’s theorem, a set *X* can only satisfy *X*≅ *X*^{X}
in the trivial cases *X*≡∅ or **1**.
Apparently, the untyped λ-calculus is therefore inconsistent.
Dana Scott eventually saw through this,
and constructed *topological lattices* with this property
[Sco93].
As these can be embedded in presheaf toposes,
where we may treat them as intuitionistic sets,
we see that Cantor’s supposedly fundamental theorem of set theory
actually relies on excluded middle.

The message that we ought to take from a counterexample
is no more than that you can’t do it *like that*.
Maybe someone someday will think of a *different* way.
They will identify underlying assumptions *A*, *B* and *C*
that had previously been overlooked,
and achieve all of *D*, *E*, *F* and *G* on some *other* basis,
using *A*′, *B*′ and *C*′ instead.

In particular, just as in the use of formal methods for security (putting a strong lock on a weak door), it is the very rigour of a fragment of argument that can be most misleading about the bigger picture. In our architectural metaphor, counterexamples are reasons for choosing one general scheme over another, whilst theorems hold the building up once we have chosen which plan to carry out. A tower block may later turn out to be a mistake for sociological reasons that have nothing to do with how accurately its plans were drawn.

It would, perhaps, be useful to make a clear distinction between
the principal track of a *theory* (from axioms to theorems) and its
*peri-theory*.
This is the discussion *around* the theory, consisting of the
main examples, prototypes and counterexamples that
*led us to the choice* of statements of the axioms and theorems.
Other examples of peritheory are converses that lead back from the
main theorem to (the necessity of) the chosen axioms,
and more generally the lists of equivalent alternative formulations
of the axioms that some textbooks like to give.

The whole of this paper (apart from §§6 & 8) concerns the peritheory of topology,

**3.4.**
Unfortunately, it is not as straightforward as we have suggested
to work from a client’s brief:
this cannot be taken literally, and has to be negotiated.

If the client is allowed to dictate what is *fundamental*,
the new foundations will be no more flexible than the old ones.
It takes an outsider’s perspective to distinguish the key elements
of an enterprise from the accumulation of detail.
The first arrow does not therefore just copy the theorems and chapter
headings of the old theory.

In approaching a mathematical discipline, category theory often
focuses on ideas that its specialists previously regarded as trivial.
For example, the subobject classifier and Sierpiński space (§7)
are merely two-element sets in classical set theory and topology.
Also, whilst mathematicians have acknowledged that
the theorems that *create* important structures are adjoints,
an adjunction is a two-way relationship.
In this, the other partner typically does something that appears to be mere
bureaucracy, namely to “forget” the same structure.

**3.5.**
Another issue on which the clients’ preconceptions need to be challenged
is the identification of their most valuable products.
It is up to their *customers*
to do this.
For example, set theorists put much of their effort into
considering infinite cardinals,
but the demand from the market is for more prosaic things,
such as quotients of equivalence relations.
These are axiomatised explicitly in categorical logic(§5).

On the other hand, maybe led by an over-emphasised counterexample,
specialists sometimes develop variant forms of a subject that
*omit* properties which the customers may regard as essential.
One case of this is compactness of the interval
[0,1]⊂ℝ (§1.8),
which is absent from Bishop’s theory [BB85].

**3.6.**
Negotiation of this kind is how we answer the final objection
to changing foundations, namely that the prevailing set-up
is crucial for the whole of science (§4).

Science and mathematics are *systems*
whose *component* disciplines interact across an *interface*.
The higher-level components depend only weakly on the details of those below,
but do have requirements for certain features.
This is a commonplace in modern technology.
It is also well known in the wider picture of science:
whilst chemistry depends on physics, it only uses four sub-atomic particles,
organic chemistry for the most part only four elements,
and genetics only four bases and twenty amino acids.
Physics very probably relies on compactness of the interval,
for example to ensure that local patches of solutions
to a differential equation fit together into a global solution,
instead of forming a singular cover (§1.8).
On the other hand, I would be very sceptical if you told me that
some property of black holes depends on excluded middle.
Have you actually developed the analogous constructive theory,
and found observational evidence to distinguish it from the classical one?
This is, after all, what the experimental method says you should do.

However, the question of whether some application in science actually depends on a particular foundational principle is never an easy one. Settling the necessity of each invocation of excluded middle, Heine–Borel or a large cardinal in the chain of theorems about functional analysis or other subjects may require a decade’s research. Unfortunately, the commonest way of handling this is “megaphone philosophy” that, if it contains any mathematics at all, consists largely of the abuse of counterexamples.

**3.7.**
When we replace a component of a system,
it needs to be backwards-compatible in its *function*,
not necessarily its *implementation* or *extent*.
After all, an architect who has been commissioned to replace a building
will only do so with an exact copy if it is to be a museum.

In the case of general topology, the methods of construction in this paper
are such that the new building must either be smaller than the old one
(consisting of just locally compact spaces) or substantially bigger.
This is entirely consistent with the historical development of the subject,
which grew from figures that are embedded in Euclidean 3-dimensional space,
to ℝ^{n}, to projective and non-Euclidean geometry,
to manifolds, to spaces of functions in analysis,
and to domains for denotational semantics of programming languages.
The intuition of continuity has been captured in numerous
quite different ways, making use of metrics, uniformity and convergence of
sequences, as well as systems of points and open neighbourhoods.
Each *axiomatisation* leads to a different *totality*
of domains of continuity.

For comparison, there can be little doubt that groups and fields
have the right axioms for the intuitions that they seek to capture.
This is indicated by the fact that we use the axioms directly
for computation.
Algebra textbooks are also able to *classify* finite fields,
and they make a serious attack on the similar (albeit intractible)
problem for groups.

The situation in general topology is much less clear. It is rather like a medieval “world” chart that more or less accurately depicts the Mediterranean, but has mythological creatures around the outside. The various approaches to continuity accurately capture real manifolds, just as the old cartographers recorded their own familiar territory, but we cannot be confident in using the outer parts of the chart.

The error in both cases is the *co-ordinate system*.
Its assumptions provide reasonable approximations locally,
but by their nature entail certain *boundaries* to the global system.
Flat charts are useful, but in ancient times Eratosthenes had known,
not only that the Earth is spherical (from lunar eclipses),
but even how big it is.
Similarly, it has been known since the 1960s that points and open sets
are the wrong co-ordinate system for topology.
Sheaves in algebraic geometry were based on open sets and not points
,
whilst algebraic topologists sought more “convenient” categories
[Bro64, Ste67],
*i.e*. those that admit general spaces of functions.
However, it is by no means clear what topologies
these function-spaces should carry,
especially if we want to invesigate the properties of
ℕ^{ℕℕ} and ℝ^{ℝℝ}(*cf*. §12.10).

There is, therefore, nothing special about the boundaries of
the category of objects that the textbooks call “topological spaces”.
These books treat non-Hausdorff spaces with derision,
and make little attempt to explore the full extent of even
the world that is measured out by their own co-ordinate system.
This was only begun when the analogy with the ∃∧-fragment
of logic was recognised, *cf*. [Vic88].
We therefore *un*define the terms “space” and “topological space”,
leaving them open to new definitions.
The textbook spaces will be re-branded as *Bourbaki spaces*
[Bou66] to strip them of their authority.

**3.8.**
Even when we have found the right co-ordinate system,
it may not be appropriate to *describe* it in the same way
for both the foundations and the applications.
Any engineer knows that the user manual for a gadget
should not be written in the same way as its technical specification.
So this paper discusses the *foundations* of ASD,
whilst [I][J]
provide separate introductions that are suitable
for its *applications* to elementary real analysis.

One reason for this is that the technical ideas may later be redeployed
to make something else with an entirely different use.
In our case, we shall find that there is a new underlying abstract structure
(§§6,12)
that has other applications besides topology,
and deserves to be studied in its own right.
It includes certain definitions that,
in the presence of the specifically topological structure,
have different characterisations (*cf*. §7.7).
This is in line with the usual experience of applying category theory
to mathematical disciplines,
namely that ideas with various different traditional formulations
turn out to be examples of a common abstract idea.

Secondly, the demands from users are often the driving forces behind advances in technology. However, as we have seen for computing hardware and software, the biggest improvements do not result from adding bells and whistles, but by re-thinking and strengthening the fundamental principles. In ASD, the principal challenge for the future is how to extend the boundaries beyond local compactness, in such a way that Banach spaces are given “the right” topology, whatever that means. In order to do this, it is the initial formulation of the underlying abstract structure in §6, rather than its adaptation to topology, that needs to be replaced.

**3.9.**
Finally,
there is an issue for which architecture is entirely the wrong metaphor
and the departure of Columbus from the Mediterranean is much more appropriate.
*Los Reyes Catolicos* certainly did not promote
free intellectual exploration in their domestic and colonial policy,
but they did at least *fund* a “blue sea” project.

Nowadays, one is asked to give advance notice of all of the theorems that
one intends to prove.
Such planning is appropriate when building a house,
but it is possible *if and only if* there are *no* original ideas.
A mathematician with a plan for a theorem wants to carry it out straight away,
and the only pieces of equipment that are needed are a clear head and
a clear blackboard.
We don’t put our *lives* at stake as Columbus did when we embark
on scientific experiments or try to prove mathematical theorems,
but if there is no *intellectual* risk of failure
in a proposed piece of research,
then it is redundant, and probably not worthy of funding.

We like to think that the *finished product* of mathematics
is the most precise of any branch of science or engineering.
The corollary of this is that the *vision* of a mathematical
project in advance of its detailed plan is necessarily *much more
vague* than in any other discipline.

And things may not go according to plan even if we do succeed, because there may be a new continent to discover. For a powerful account that has a far wider relevance than to physics, see Part IV of [Smo06], which demonstrates a familiarity with the real experience of those who do revolutionary science that is absent from [Kuh62].

It is time to apply this method to some mathematics.