Ideologists (a term used in IC to mean students of ideology) have to be interested in enquiries into the sources from which knowledge is derived, and in his essay entitled “Logic, Mathematics and Knowledge of Nature,”  Hans Hahn distinguishes two possible ones: observation and thinking. Comments are to be made, but first some of the main points of the article. (With a recommendation that it be read as a whole).
Perceiving that observation can be deceptive the ancients came to prefer thinking as a source of knowledge. But the failure of rationalism to produce anything of value, together with the great success achieved by modern science, relying largely upon careful observation, caused a shift towards empiricism.
Knowledge based upon observation can never be absolutely reliable; every scientific law, no matter how well founded, remains always at the mercy of future observations which may invalidate it. The observation, for example, that Mercury did not appear at a particular place and time led to the abandonment of Newtonian gravitation in favour of Einsteinian.
Mathematics does not suffer from this limitation. Two and three make five, and cannot fail to do so, because mathematics does no more than develop the implications of the rules with which it begins; 2 + 3 must always be equal to 5 because they are no more than different ways of saying the same thing, the statement 2 + 3 = 5 is a tautology.
Similarly with logic. If the terms snow rose and helleborus niger both be adopted as names for plants of a given species we can say, with absolute certainty, that every snow rose is a helleborus niger. No observation can ever disprove this proposition. In a similar way we decide to designate some coloured objects as red and stipulate that all other objects be designated not red. Having done this we can assert with absolute certainty: firstly, that nothing is both red and not red (the law of contradiction); secondly, that everything is either red or not red (the law of the excluded middle). These propositions, also, are incapable of falsification by observation since they have not been derived from observation but from definition.
Every time we formulate a hypothesis, even if it be one belonging to that special class known as laws of nature, we implicitly assert many other propositions, and the function of logic and mathematics is to bring these out. This is their only function; they do not provide information about the real world.
The validity of a law of nature, however, is a different thing from its implications, and depends not upon logical or mathematical demonstration but upon confirmation by observation. These laws are therefore subject to falsification by further observation (as happened with Newton’s law of gravitation).
On the one side logic and mathematics, incontrovertible (in the technical phrase apodeictically certain) but telling us nothing we did not know, implicitly, beforehand. On the other side hypotheses, some possessing the status of laws of nature, organising information about the real world but always liable to be invalidated by future observations.
This paper appeared in a collection of essays on logical positivism edited by A. J. Ayer, and the emphasis on sharp distinction running through the book provides outstanding examples of the ideology of precision at work. As a stimulus to clear thinking and a protection against waffle the approach is invaluable. It invites us to apply to these essays the sharp criticism they use themselves, and when we do this it soon turns out that valuable as Hahn’s argument may be for the limited purposes of physical science it falls something short of completely solving the problem it tackles.
Hahn tells us directly: “observation is the only source of knowledge of facts.” But he also shows that knowledge, as it is used in science and elsewhere, does not consist only of facts. In his own words: “A statement which really says something about the objects which it mentions, is the following: ‘If you heat this piece of iron up to 800° it will turn red… ‘” That does not rest upon the observations that this, that and the other piece of iron turned red when heated, for those give no warrant for statements about other pieces of iron. The quoted statement is linked to the observations by the hypothesis that every piece of iron, when heated to that degree, will turn red. This is no isolated example Hahn speaks repeatedly of hypotheses, and although these may be given logical or mathematical form, (for example the laws of nature formulated by science) they are not purely logical or mathematical formulae; they relate to the real world.
But they are not themselves observable:
Observation discloses to me only the transient, it does not reach beyond the observed; there is no bond that would lead from one observed fact to another…
Nobody has ever seen, or ever will see, hear, feel or taste gravitation or the conservation of energy. We perceive events – apples falling, planets moving, heat being produced by friction – and in order to increase our ability to handle the world we live in, to avoid being caught by surprise the next time one of these events occurs, we hypothesise regularity, sometimes achieving the reliability of Newton’s law of gravitation and the second law of thermodynamics. But this regularity is never observed; it is just as much a mental creation as the “forces,” “inherent tendencies” or “natural inclinations” that satisfy the casual thinking of everyday life. The question arises: Where do these hypotheses come from?
They are not produced by logic or mathematics, for the whole burden of Hahn’s treatment of these is to show that their function is restricted to developing the implications lying dormant in any hypothesis; the hypothesis comes first, mathematics and logic working upon it later.
Neither are they among the results of observation. A hypothesis may deal with these, but it goes beyond them, and as Hahn has said, observation provides no bond leading from one observed fact to another. The law of gravitation, for example, speaks of all mass points, and Hahn tells us “nobody can observe all mass points.”
The source from which hypotheses spring begins to appear when we note that radically different hypotheses may be formulated to deal with similar sets of observations. Observations of the change in colour when pieces of metal are heated may be dealt with by hypothesising that the spirit in the metal is displaying its anger at such treatment, that God arranged things that way, or that the explanation advanced by physical science is the correct one. Scientists themselves accept different hypotheses according to the circumstances; for many purposes they continue to use Newtonian gravitation (indeed, as Kuhn has pointed out, they still live for some purposes in a Ptolemaic rather than a Copernican universe).
And, again as Hahn has shown, although the hypotheses of physical science may be better supported than the others, it is not the case that they are right and the others wrong, even the law of gravitation, the second law of thermodynamics and E = MC2 remain only provisionally established, subject to falsification by further observation.
Hypotheses comprise results of observation but go beyond them; they arise from interaction between observation and the mental activity of the observer, and this mental activity is governed by the observer’s pre-existing assumptions. Hypotheses belong among ideological phenomena, and since Hahn has shown that they include the laws of science, we have to accept that science, even physical science, is an ideological activity.
The phrase ‘laws of nature’ is generally used only for the imposing formulations advanced in the name of science, but in showing these to be no more than hypotheses (though unusually well-supported ones) Hahn reveals this limitation as arbitrary. In everyday life we are constantly making observations and forming hypotheses about them, and the difference between ‘fire burns’ and the second law of thermodynamics is one of degree. The achievements of modern science are great, but the achievements of the unscientific, unprincipled, expedient mode of thought which enabled the race to persist over some two million years, and to occupy the planet, are at least equal in importance. These provide most of the base on which science rests, and while they have demonstrated their ability to subsist without science it would be helpless without them.
But science presupposes more than expedience. It requires also supplies, facilities, equipment, security, educated people, and an organised society with a tradition of respect for intellectual endeavour. That is to say, it depends also upon the presence of the domination / submission ideology as a significant social influence.
 In Ayer A. J. Ed. 1963 Logical Positivism Glencoe: The Free Press of Glencoe.
from Ideological Commentary 44, March 1990.