We see that experience plays an indispensable role in the genesis of geometry; but it would be an error thence to conclude that geometry is, even in part, an experimental science. If it were experimental it would be only approximative and provisional. And what rough approximation! ...The object of geometry is the study of a particular 'group'; but the general group concept pre-exists... in our minds. It is imposed on us, not as form of our sense, but as form of our understanding. Only, from among all the possible groups, that must be chosen... will be... the standard to which we shall refer natural phenomena. Experience guides us in this choice without forcing it upon us; it tells us not which is the truest geometry, but which is the most convenient. Notice that I have been able to describe the fantastic worlds... imagined without ceasing to employ the language of ordinary geometry.
Henri PoincaréRelated topics
able approximation ceasing choice concept employ error experience experimental fantastic form general genesis geometry language natural notice possible science see sense study understandingRelated quotes
Contemplating the universe, the whole system of creation, in this point of light, we shall discover, that all that which is called natural philosophy is properly a divine study- It is the study of God through his works - It is the best study, by which we can arrive at a knowledge of the existence, and the only one by which we can gain a glimpse of his perfection.
Do we want to contemplate his power? We see it in the immensity of the Creation. Do we want to contemplate his wisdom? We see it in the unchangeable order by which the incomprehensible Whole is governed. Do we want to contemplate his munificence? We see it in the abundance with which he fills the earth. Do we want to contemplate his mercy? We see it in his not withholding that abundance even from the unthankful. In fine, do we want to know what God is? Search not written or printed books, but the Scripture called the Creation.
Thomas Paine
Carbon dioxide, Mister Speaker, is a natural byproduct of nature. Carbon dioxide is natural. It occurs in Earth. It is a part of the regular lifecycle of Earth. In fact, life on planet Earth can't even exist without carbon dioxide. So necessary is it to human life, to animal life, to plant life, to the oceans, to the vegetation that's on the Earth, to the, to the fowl that - that flies in the air, we need to have carbon dioxide as part of the fundamental lifecycle of Earth...There isn't one such study because carbon dioxide is not a harmful gas, it is a harmless gas. Carbon dioxide is natural. It is not harmful. It is part of Earth's life cycle...And yet we're being told that we have to reduce this natural substance and reduce the American standard of living to create an arbitrary reduction in something that is naturally occuring in the earth.
Michele Bachmann
The use of the intellect in the sciences whose primitive concepts as well as axioms are given by sensuous intuition is only logical, that is, by it we only subordinate cognitions to one another according to their relative universality conformably to the principle of contradiction, phenomena to more general phenomena, and consequences of pure intuition to intuitive axioms. But in pure philosophy, such as metaphysics, in which the use of the intellect in respect to principles is real, that is to say, where the primary concept of things and relations and the very axioms are given originally by the pure intellect itself, and not being intuitions do not enjoy immunity from error, the method precedes the whole science, and whatever is attempted before its precepts are thoroughly discussed and firmly established is looked upon as rashly conceived and to be rejected among vain instances of mental playfulness.
Immanuel Kant
At the point at which the concept of différance, and the chain attached to it, intervenes, all the conceptual oppositions of metaphysics (signifier/signified; sensible/intelligible; writing/speech; passivity/activity; etc.)- to the extent that they ultimately refer to the presence of something present (for example, in the form of the identity of the subject who is present for all his operations, present beneath every accident or event, self-present in its "living speech," in its enunciations, in the present objects and acts of its language, etc.)- become non pertinent. They all amount, at one moment or another, to a subordination of the movement of différance in favor of the presence of a value or a meaning supposedly antecedent to différance, more original than it, exceeding and governing it in the last analysis. This is still the presence of what we called above the "transcendental signified.
Jacques Derrida
Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.
William Kingdon Clifford
