Yet this failure to agree on semantics is in part because of the difficulty in providing solid, incontrovertible justifications for these meanings. Consider the new branches of mathematics that were [created|discovered] when Euclid's definition of a straight line were questioned.
That mathematics corresponds so well to the world we perceive is amazing. Why should this be the case? How can we be sure that mathematics and science holds for all cases which we do not observe or that they will continue to do so? Can rigorous justifications be given for these questions that do not rely on circular arguments and blind faith?
Above you say that this is an honest request to know more so I will treat it as such.
In philosophy a method proposed by Descartes to find the foundations of knowledge was to doubt everything that possibly could be doubted which includes the external world. Wittgenstein in his Tractatus similarly starts from a blank slate and tries to define and describe the world without crossing his own boundaries and definitions of what can be considered sensible to say (and fails. To objectively describe the world is an attempt to step outside it and his bounds of sense).
I would also like to note that this early is work very different from his later work where he completely rejected the Tractatus, writing with a different style and focus. His later work (especially Philosophical Investigations) has some extremely interesting ideas regarding language, its use and development which I believe would interest those working in the areas of semantic web and NLP.
<cheap shot>And what is this substance of mathematics?</cheap shot>
Calling mathematics "incontrovertibly true" and stating that abstractions used in mathematics can be explained concretely in simpler terms are philosophical positions and the justifications for and against these positions are debated outside mathematics in philosophy. You praise the rigor of mathematics while ignoring the rigor in which philosophy searches for the justifications of assumptions for all fields of human knowledge including mathematics and science.
That mathematics corresponds so well to the world we perceive is amazing. Why should this be the case? How can we be sure that mathematics and science holds for all cases which we do not observe or that they will continue to do so? Can rigorous justifications be given for these questions that do not rely on circular arguments and blind faith?