scholarly journals HERBRAND’S THEOREM AND NON-EUCLIDEAN GEOMETRY

2015 ◽  
Vol 21 (2) ◽  
pp. 111-122 ◽  
Author(s):  
MICHAEL BEESON ◽  
PIERRE BOUTRY ◽  
JULIEN NARBOUX
Keyword(s):  
Euclidean Geometry ◽  
The Other ◽  
Basic Theorem ◽  
Very Old ◽  
First Order ◽  
Herbrand’S Theorem ◽  
Non Euclidean Geometry ◽  
Herbrand's Theorem

AbstractWe use Herbrand’s theorem to give a new proof that Euclid’s parallel axiom is not derivable from the other axioms of first-order Euclidean geometry. Previous proofs involve constructing models of non-Euclidean geometry. This proof uses a very old and basic theorem of logic together with some simple properties of ruler-and-compass constructions to give a short, simple, and intuitively appealing proof.

Philosophy and Non-Euclidean Geometry

1919 ◽  
Vol 11 (4) ◽  
pp. 196-198
Author(s):  
F. A. Foraker
Keyword(s):  
Euclidean Geometry ◽  
History Of Philosophy ◽  
The Other ◽  
High Rank ◽  
Non Euclidean Geometry ◽  
History Of ◽  
And Mathematics ◽  
A Minor

Leibnitz and Descartes made remarkable contributions to both mathematics and philosophy. Newton obtains a high rank in the history of the former subject, but only a minor place in the history of philosophy, while Kant, who possessed a well-founded knowledge of the science and mathematics of his time, receives one of the foremost positions in the history of philosophy. Upon the basis of these facts, if we neglect a few of the lesser lights, the statement is often made that there is a relationship between the study of mathematics and the study of philosophy, and that he who studies one of them will also find himself a devotee in the pursuit of the other.

The Emergence of Foundational Study

2017 ◽  
Author(s):  
Jan von Plato
Keyword(s):  
High School ◽  
Euclidean Geometry ◽  
School Teacher ◽  
High School Teacher ◽  
The Other ◽  
Non Euclidean Geometry ◽  
Higher Institutions ◽  
Hermann Graßmann

This chapter looks at how modern foundational study has twofold mathematical roots. One is the discovery of non-Euclidean geometries, especially the proof of independence of the parallel postulate by Eugenio Beltrami in 1868, in his Saggio di interpretazione della geometria non-euclidea (Treatise on the interpretation of non-Euclidean geometry). The other root is arithmetical, retraceable through Peano and others to the 1861 book Lehrbuch der Arithmetik für höhere Lehranstalten (Arithmetic for higher institutions of learning) by the high school teacher Hermann Grassmann. In each of these two cases, one has to set things straight: To prove independence in geometry, one has to ask what the axioms are, and maybe even the principles of proof.

A strong version of Herbrand's theorem for introvert sentences

1998 ◽  
Vol 63 (2) ◽  
pp. 555-569 ◽  
Author(s):  
Tore Langholm
Keyword(s):  
Ground Term ◽  
New Members ◽  
First Order ◽  
Herbrand’S Theorem ◽  
Universal Sentence ◽  
Order Language ◽  
The Matrix ◽  
Image Position ◽  
Infinite Set ◽  
Herbrand's Theorem

A version of Herbrand's theorem tells us that a universal sentence of a first-order language with at least one constant is satisfiable if and only if the conjunction of all its ground instances is. In general the set of such instances is infinite, and arbitrarily large finite subsets may have to be inspected in order to detect inconsistency. Essentially, the reason that every member of such an infinite set may potentially matter, can be traced back to sentences like(1) Loosely put, such sentences effectively sabotage any attempt to build a model from below in a finite number of steps, since new members of the Herbrand universe are constantly brought to attention. Since they cause an indefinite expansion of the relevant part of the Herbrand universe, such sentences could quite appropriately be called expanding.When such sentences are banned, stronger versions of Herbrand's theorem can be stated. Define a clause (disjunction of literals) to be non-expanding if every non-ground term occurring in a positive literal also occurs (possibly as an embedded subterm) in a negative literal of the same clause. Written as a disjunction of literals, the matrix of (1) clearly fails this criterion. Moreover, say that a sentence is non-expanding if it is a universal sentence with a quantifier-free matrix that is a conjunction of non-expanding clauses. Such sentences do in a sense never reach out beyond themselves, and the relevant part of the Herbrand universe is therefore drastically reduced.

Linear reasoning. A new form of the Herbrand-Gentzen theorem

1957 ◽  
Vol 22 (3) ◽  
pp. 250-268 ◽  
Author(s):  
William Craig
Keyword(s):  
Normal Form ◽  
Distinctive Feature ◽  
First Order ◽  
Herbrand’S Theorem ◽  
Free Level ◽  
New Form ◽  
The Relationship ◽  
Herbrand's Theorem

In Herbrand's Theorem [2] or Gentzen's Extended Hauptsatz [1], a certain relationship is asserted to hold between the structures of A and A′, whenever A implies A′ (i.e., A ⊃ A′ is valid) and moreover A is a conjunction and A′ an alternation of first-order formulas in prenex normal form. Unfortunately, the relationship is described in a roundabout way, by relating A and A′ to a quantifier-free tautology. One purpose of this paper is to provide a description which in certain respects is more direct. Roughly speaking, ascent to A ⊃ A′ from a quantifier-free level will be replaced by movement from A to A′ on the quantificational level. Each movement will be closely related to the ascent it replaces.The new description makes use of a set L of rules of inference, the L-rules. L is complete in the sense that, if A is a conjunction and A′ an alternation of first-order formulas in prenex normal form, and if A ⊃ A′ is valid, then A′ can be obtained from A by an L-deduction, i.e., by applications of L-rules only. The distinctive feature of L is that each L-rule possesses two characteristics which, especially in combination, are desirable. First, each L-rule yields only conclusions implied by the premisses.

scholarly journals Thematic Maps for the Variation of Bearing Capacity of Soil Using SPTs and MATLAB

Geosciences ◽  
2020 ◽  
Vol 10 (9) ◽  
pp. 329
Author(s):  
Mahdi O. Karkush ◽  
Mahmood D. Ahmed ◽  
Ammar Abdul-Hassan Sheikha ◽  
Ayad Al-Rumaithi
Keyword(s):  
Bearing Capacity ◽  
Mean Squared Error ◽  
Ground Level ◽  
The Other ◽  
Thematic Maps ◽  
First Order ◽  
Squared Error ◽  
Order Polynomial ◽  
Interpolation Polynomials ◽  
Penetration Tests

The current study involves placing 135 boreholes drilled to a depth of 10 m below the existing ground level. Three standard penetration tests (SPT) are performed at depths of 1.5, 6, and 9.5 m for each borehole. To produce thematic maps with coordinates and depths for the bearing capacity variation of the soil, a numerical analysis was conducted using MATLAB software. Despite several-order interpolation polynomials being used to estimate the bearing capacity of soil, the first-order polynomial was the best among the other trials due to its simplicity and fast calculations. Additionally, the root mean squared error (RMSE) was almost the same for the all of the tried models. The results of the study can be summarized by the production of thematic maps showing the variation of the bearing capacity of the soil over the whole area of Al-Basrah city correlated with several depths. The bearing capacity of soil obtained from the suggested first-order polynomial matches well with those calculated from the results of SPTs with a deviation of ±30% at a 95% confidence interval.

scholarly journals “I have Learned Non-Euclidean Geometry Just from this Book” - Some facets of the correspondence between Friedrich Engel and David Hilbert

2021 ◽  
Vol 76 (3) ◽  
Author(s):  
Peter Ullrich
Keyword(s):  
Present Article ◽  
19Th Century ◽  
Euclidean Geometry ◽  
Academic Life ◽  
David Hilbert ◽  
Non Euclidean Geometry ◽  
The 19Th Century

AbstractFriedrich Engel and David Hilbert learned to know each other at Leipzig in 1885 and exchanged letters in particular during the next 15 years which contain interesting information on the academic life of mathematicians at the end of the 19th century. In the present article we will mainly discuss a statement by Hilbert himself on Moritz Pasch’s influence on his views of geometry, and on personnel politics concerning Hermann Minkowski and Eduard Study but also Engel himself.

scholarly journals Langevin simulations of protoplasmic streaming in non-Euclidean geometry

2021 ◽  
Vol 1730 (1) ◽  
pp. 012037
Author(s):  
Shuta Noro ◽  
Masahiko Okumura ◽  
Satoshi Hongo ◽  
Shinichiro Nagahiro ◽  
Toshiyuki Ikai ◽  
...  
Keyword(s):  
Euclidean Geometry ◽  
Protoplasmic Streaming ◽  
Non Euclidean Geometry
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