Moreover, it shows us, at least, that things like the natural numbers can be conjured literally out of nothing, merely by employing the abstract notion of ‘set’. We get an infinite sequence of abstract (Platonic) mathematical entities — sets containing, respectively, zero, one, two, three, etc., elements, one set for each of the natural numbers, quite independently of the actual physical nature of the universe. In Fig.1.3 we envisaged a kind of independent ‘existence’ for Platonic mathematical notions — in this case, the natural numbers themselves — yet this ‘existence’ can seemingly be conjured up by, and certainly accessed by, the mere exercise of our mental imaginations, without any reference to the details of the nature of the physical universe. Dedekind’s construction, moreover, shows how this ‘purely mental’ kind of procedure can be carried further, enabling us to ‘construct’ the entire system of real numbers, still without any reference to the actual physical nature of the world. Yet, as indicated above, ‘real numbers’ indeed seem to have a direct relevance to the real structure of the world — illustrating the very mysterious nature of the ‘first mystery’ depicted in Fig.1.3.
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From the perspective of Systemic Functional Linguistic Theory, the 'physical nature of the world' is first-order meaning, construed of experience. The 'construction' of 'the entire system of real numbers' is the reconstrual of first-order meanings, cardinal numbers, as the second-order meanings of mathematics. This is why the real numbers of mathematics have 'direct relevance' to the structure of the physical world.
The expansion of the system of mathematics is achieved through instances of the system, which may be mentally projected as ideas, or verbally projected as locutions (spoken or written texts). It is only the latter that is shared between mathematicians, and which, therefore, expands the shared potential of the mathematical community.
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