In order to understand the opening statements in Tractatus it is necessary to get inside the mind of Wittgenstein and see things how he was seeing them. The foundations of his theory of meaning were highly axiomatic (self-evident) in nature. He was laying down early on what he thought simply must be the case if we, as it seems, are able to represent elements of the world with the language we use. In this early part of Tractatus where he lays the foundations of his theory, he says virtually nothing (or at least provides no elucidation) of details of the theory which weren’t strictly part of this rock solid scaffolding upon which his theory was built. He was stipulating what in his view just has to be the case given the way language seems to mirror reality. If we represent the world through language, in Wittgenstein’s view, then certain strict conditions have to be met. A failure to understand this crucial point at the outset is nothing short of failing to understand the foundations of Tractatus itself, at least in the fullest sense.
His vagueness on certain details has always been a source of confusion amongst philosophers, who sometimes (erroneously) fill in the missing details with what seems like reasonably intuitive proposals. It is not altogether clear what Wittgenstein’s views are on some of these missing details, or even if he was certain himself. In fact it was an important part of his theory that certain things are necessarily hidden from us. But I certainly believe the concise and conservative assertions made in the early part of Tractatus were partly Wittgenstein’s attempts to establish and make evident his axiomatic approach to the problem, and moreover to make a crystal clear distinction between what was necessarily true and what could only (perhaps reasonably) be surmised. Why he wasn’t more explicit with regards to this I am not sure, although it did seem to be part of his nature not to always make things explicit. But nonetheless this is unquestionably what he was up to.
To explain what an axiom is consider the following statements:-
All fish swim in water
Goldie is a fish
Therefore Goldie swims in water
The first two lines are premises or assumptions. The final line is the conclusion, and is simply a logical consequence of the two assumptions being made. The assumptions are in essence acting as axioms and the final line is essentially a theorem. In a similiar manner the whole of Euclidean geometry is a logical consequence of 5 basic assumptions, or self evident facts. It is nothing more than a set of theorems derived from 5 axioms. The principle is exactly the same, only the details differ. The logical arguments are not generally of this particular form of course, and tend to involve more subtle reasoning, but the process of deductively proving theorems from assumptions is exactly the same. It is really a very straight forward concept. There is nothing deep about it. Axioms are assumptions and theorems are simply logical consequences of axioms. (You may find it hard to believe that entire areas of mathematics such as geometry can be built up from just a few underlying assumptions, but once you have established some theorems these can in turn be used as building blocks themselves for further theorems, which in turn become building blocks themselves, and so on). Interestingly if you drop one of the five axioms of Euclidean geometry, the one that states that parallel straight lines never meet, you end up with non-Euclidean geometry. (Strictly speaking the axiom isn't dropped. It's simply changed from parallel lines never meet to parallel lines will meet if they are long enough).
Other fields of mathematics are built using exactly this same process of deductive logic. However the axioms of pure mathematics can never be incorrect (although they could be inappropriate). They are simply your starting point upon which to build your theorems. The example of Euclidean geometry provides a perfect example of this. The fact that altering one axiom leads to non-Euclidean geometry doesn’t make the original axiom incorrect. It’s modification simply leads to a different set of theorems. Of course the theorems themselves could potentially be incorrect if you fail to use the logic properly.
In the case of axioms relating to physical sciences things are different. Any axiom making an assertion about the physical world could always be incorrect in principle. However in practise this rarely happens because they are normally very carefully selected in the first place. They are meant to be self-evident truths (as far as possible) which require no justification for believing. They simply stand on their own merit without any further requirement to be validated. In deed there is usually no way to validate them, at least on strictly logical grounds, although they will generally gain considerable credibility by the predictive power of the resulting theorems based on them. In science the axioms are typically nothing more than the formal declaration of direct empirical observations (Newtonian mechanics being a very good example) whereas in philosophy the beginning assumptions can be slightly more open to debate. Wittgenstein effectively purports his assertions at the beginning of Tractatus to be self-evident, at least after some period of reflection.
Wittgenstein’s theory doesn’t necessarily entail we live solipsistically in our own private worlds, but it is certainly designed to accommodate that view. And the spirit of this idea runs the entire way through Tractatus in as far as the very essence of the theory is attempting to explain how worlds that are subjectively and qualitatively different from each other (within some well defined constraints) can nevertheless be isomorphically mapped onto each other. It is this isomorphic mapping that provides the crucial link which makes communication possible. I explain all this in extremely clear terms later on.
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