Chuck Darwin<p>We are now concerned with more radical possibilities. </p><p>A paradigmatic example is topology. </p><p>In modern “analytic topology”, a “space” is defined to be a set of points equipped with a collection of subsets called open, <br>which describe how the points vary continuously into each other.<br> (Most analytic topologists, being unaware of synthetic topology, would call their subject simply “topology.”) </p><p>By contrast, in synthetic topology we postulate instead an axiomatic theory, on the same ontological level as ZFC, <br>whose basic objects are spaces rather than sets.</p><p>Of course, by saying that the basic objects “are” spaces we do not mean that they are sets equipped with open subsets. </p><p>Instead we mean that “space” is an undefined word, <br>and the rules of the theory cause these “spaces” to behave more or less like we expect spaces to behave. </p><p>In particular, synthetic spaces have open subsets (or, more accurately, open subspaces), <br>but they are not defined by specifying a set together with a collection of open subsets.</p><p>It turns out that synthetic topology, like synthetic set theory (ZFC), is rich enough to encode all of mathematics. </p><p>There is one trivial sense in which this is true: <br>among all analytic spaces we find the subclass of indiscrete ones, <br>in which the only open subsets are the empty set and the whole space. </p><p>A notion of “indiscrete space” can also be defined in synthetic topology, <br>and the collection of such spaces forms a universe of ETCS-like sets <br>(we’ll come back to these in later installments). </p><p>Thus we could use them to encode mathematics, entirely ignoring the rest of the synthetic theory of spaces. <br>(The same could be said about the discrete spaces, <br>in which every subset is open; <br>but these are harder (though not impossible) to define and work with synthetically. </p><p>The relation between the discrete and indiscrete spaces, <br>and how they sit inside the synthetic theory of spaces, <br>is central to the synthetic theory of cohesion, <br>which I believe David is going to mention in his chapter about the philosophy of geometry.)</p><p>However, a less boring approach is to construct the objects of mathematics directly as spaces. </p><p>How does this work? <br>It turns out that the basic constructions on sets that we use to build (say) the set of real numbers have close analogues that act on spaces. </p><p>Thus, in synthetic topology we can use these constructions to build the space of real numbers directly. </p><p>If our system of synthetic topology is set up well, <br>then the resulting space will behave like the analytic space of real numbers<br> (the one that is defined by first constructing the mere set of real numbers and then equipping it with the unions of open intervals as its topology).</p><p>The next question is, <br>why would we want to do mathematics this way? </p><p>There are a lot of reasons, <br>but right now I believe they can be classified into three sorts: <br>modularity, <br>philosophy, and <br>pragmatism. </p><p>(If you can think of other reasons that I’m forgetting, please mention them in the comments!)</p><p>By “<a href="https://c.im/tags/modularity" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>modularity</span></a>” I mean the same thing as does a programmer: </p><p>even if we believe that spaces are ultimately built analytically out of sets, <br>it is often useful to isolate their fundamental properties and work with those abstractly. </p><p>One advantage of this is <a href="https://c.im/tags/generality" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>generality</span></a>. <br>For instance, any theorem proven in Euclid’s “neutral geometry” <br>(i.e. without using the parallel postulate) <br>is true not only in the model of ordered pairs of real numbers, <br>but also in the various non-Euclidean geometries. </p><p>Similarly, a theorem proven in synthetic topology may be true not only about ordinary topological spaces, <br>but also about other variant theories such as topological sheaves, smooth spaces, etc. </p><p>As always in mathematics, if we state only the assumptions we need, our theorems become more general.</p><p> <a href="https://c.im/tags/analytic" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>analytic</span></a> <a href="https://c.im/tags/synthetic" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>synthetic</span></a></p>