2. One might think of ‘empirical space’ as something that must be measured. The idea is: ‘If it cannot be measured, then it is not relevant.’ OK, this seems fine in principle. But if a physicist would use ‘light’ as a measuring rod, then this is asking for problems. Namely, Euclidean geometry already provides us with our system of measurement. Defining ‘empirical space’ differently would conflict with our original definitional grasp of space. Better is: to stick to the definition, and regard measurements that deviate - e.g. from gravitational deflection - as the physical properties of the objects and measurement tools involved.
3. That there is a difference between mathematics and science does not disqualify the notion of absolute truth. A true deductive sequence ‘Assumption
Conclusion’ has absolute truth. And it should be realised that scientific theories are mathematical (with the scientist working on an assumption).
4. It is possible to translate the Dutch ‘lijn’ as ‘point’, and ‘punt’ as ‘line’ (thus conversely) and still find a consistent model for Euclid’s axioms. But this is a mathematical exercise, and it does not necessarily have to do with ‘space’.
So it seems that Barrow and I agree for 99%, but still, the 1% difference features big in some dimension. Note that the discussion here concerns more a side issue, but it remains useful to indicate the deeper aspects of Pythagoras’s theorem.
Falsification
The ‘principle of falsification’ is that hypotheses are only scientific if they are formulated such that they are vulnerable to empirical testing, and might be falsified. It has been formulated by Popper, see Keuzenkamp (1994).
The principle has two disadvantages: (1) purely logical, (2) stochastically.
(ad 1) Take logic first.