A quotes list created by Lee Sonogan
Wanting to do something with logic more than Aristotle, his first incompleteness theorem was published when he was 25 years old. Then his axiom of choice and continuum hypothesis investigated the foundations of mathematics as a whole. His proof theory techniques crossed into the classical, intuitionistic and modal thinking of modern times. Last a side note of his life and death from 1906 to 1978.
- The more I think about language, the more it amazes me that people ever understand each other at all.
- A consistency proof for [any] system … can be carried out only by means of modes of inference that are not formalized in the system … itself.
- Either mathematics is too big for the human mind or the human mind is more than a machine.
- All generalisations – perhaps except this one – are false.
- I don’t believe in empirical science. I only believe in a priori truth.
- I am convinced of the afterlife, independent of theology. If the world is rationally constructed, there must be an afterlife
- But every error is due to extraneous factors (such as emotion and education); reason itself does not err.
- The physical laws, in their observable consequences, have a finite limit of precision.
- Ninety percent of [contemporary philosophers] see their principle task as that of beating religion out of men’s heads. … We are far from being able to provide scientific basis for the theological world view.
- …a consistency proof for [any] system … can be carried out only by means of modes of inference that are not formalized in the system … itself.
- The meaning of world is the separation of wish and fact.
- Non-standard analysis frequently simplifies substantially the proofs, not only of elementary theorems, but also of deep results. This is true, e.g., also for the proof of the existence of invariant subspaces for compact operators, disregarding the improvement of the result; and it is true in an even higher degree in other cases. This state of affairs should prevent a rather common misinterpretation of non-standard analysis, namely the idea that it is some kind of extravagance or fad of mathematical logicians. Nothing could be farther from the truth. Rather, there are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future.
- Classes and concepts may, however, also be conceived as real objects, namely classes as “pluralities of things” or as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions. It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions…
- The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules… One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms.
- The formation in geological time of the human body by the laws of physics (or any other laws of similar nature), starting from a random distribution of elementary particles and the field is as unlikely as the separation of the atmosphere into its components. The complexity of the living things has to be present within the material, from which they are derived, or in the laws, governing their formation.
In both social and natural sciences, the body of positive knowledge grows by the failure of a tentative hypothesis to predict phenomena the hypothesis professes to explain; by the patching up of that hypothesis until someone suggests a new hypothesis that more elegantly or simply embodies the troublesome phenomena, and so on ad infinitum. In both, experiment is sometimes possible, sometimes not (witness meteorology). In both, no experiment is ever completely controlled, and experience often offers evidence that is the equivalent of controlled experiment. In both, there is no way to have a self-contained closed system or to avoid interaction between the observer and the observed. The Gödel theorem in mathematics, the Heisenberg uncertainty principle in physics, the self-fulfilling or self-defeating prophecy in the social sciences all exemplify these limitations. — Milton Friedman