The present book has been written by two mathematicians and one physicist: a pure mathematician specializing in Finsler geometry (Makoto Matsumoto), one working in mathematical biology (Peter Antonelli), and a mathematical physicist specializing in information thermodynamics (Roman Ingarden). The main purpose of this book is to present the principles and methods of sprays (path spaces) and Finsler spaces together with examples of applications to physical and life sciences. It is our aim to write an introductory book on Finsler geometry and its applications at a fairly advanced level. It is intended especially for graduate students in pure mathemat­ ics, science and applied mathematics, but should be also of interest to those pure "Finslerists" who would like to see their subject applied. After more than 70 years of relatively slow development Finsler geometry is now a modern subject with a large body of theorems and techniques and has math­ ematical content comparable to any field of modern differential geometry. The time has come to say this in full voice, against those who have thought Finsler geometry, because of its computational complexity, is only of marginal interest and with prac­ tically no interesting applications. Contrary to these outdated fossilized opinions, we believe "the world is Finslerian" in a true sense and we will try to show this in our application in thermodynamics, optics, ecology, evolution and developmental biology. On the other hand, while the complexity of the subject has not disappeared, the modern bundle theoretic approach has increased greatly its understandability.

This book has a long history of more than 20 years. The first attempt to write a monograph on information-theoretic approach to thermodynamics was done by one of the authors (RSI) in 1974 when he published, in the preprint form, two volumes of the book "Information Theory and Thermodynamics" concerning classical and quantum information theory, [153] (220 pp.), [154] (185 pp.). In spite of the encouraging remarks by some of the readers, the physical part of this book was never written except for the first chapter. Now this material is written completely anew and in much greater extent. A few years earlier, in 1970, second author of the present book, (AK), a doctoral student and collaborator of RSI in Toruli, published in Polish, also as a preprint, his habilitation dissertation "Information-theoretical decision scheme in quantum statistical mechanics" [196] (96 pp.). This small monograph presented his original results in the physical part of the theory developed in the Torun school. Unfortunately, this preprint was never published in English. The present book contains all these results in a much more modern and developed form.

Roman Ingarden studied under Husserl before and during the first world war. He belonged to the so-called Gottingen group of Husserl's pupils. Husserl's doctrine was accepted by them and interpreted in a realist vein. Ingarden defended this view all his life. He opposed the development of phenomenology towards idealism. A considerable part of Ingarden's great creative effort is dedicated to the construction of a realist phenomenology and thus, according to him, to continuing the erection of the theoret­ ical structure whose foundations were laid by Husserl in his Logical Investigations. From Ingarden's standpoint the question of idealism versus realism was a crucial one. Ingarden published several studies on Husserl. The first one was written in 1918 and the last one was published posthumously. The present essay was printed in Ingarden's book Z badan nad filozofi:t­ wsp61czesn:t- (Inquiries into Contemporary Philosophy 1963) along with a number of other essays on Husserl and his philoso­ phy. This one is representative for Ingarden's positions. It is a good example of his contribution to an important controversy in the history of phenomenology, and it gives the reader an idea of Ingarden's critique of Husserlian idealism against the background of his argument for realism. Thanks and acknowledgements are due to Mr. J. E. Llewelyn of Edinburgh University. This translation was undertaken in collaboration with him. Arn6r Hannibalsson K6pavogur, Iceland 2I. II.

The present book has been written by two mathematicians and one physicist: a pure mathematician specializing in Finsler geometry (Makoto Matsumoto), one working in mathematical biology (Peter Antonelli), and a mathematical physicist specializing in information thermodynamics (Roman Ingarden). The main purpose of this book is to present the principles and methods of sprays (path spaces) and Finsler spaces together with examples of applications to physical and life sciences. It is our aim to write an introductory book on Finsler geometry and its applications at a fairly advanced level. It is intended especially for graduate students in pure mathemat­ ics, science and applied mathematics, but should be also of interest to those pure "Finslerists" who would like to see their subject applied. After more than 70 years of relatively slow development Finsler geometry is now a modern subject with a large body of theorems and techniques and has math­ ematical content comparable to any field of modern differential geometry. The time has come to say this in full voice, against those who have thought Finsler geometry, because of its computational complexity, is only of marginal interest and with prac­ tically no interesting applications. Contrary to these outdated fossilized opinions, we believe "the world is Finslerian" in a true sense and we will try to show this in our application in thermodynamics, optics, ecology, evolution and developmental biology. On the other hand, while the complexity of the subject has not disappeared, the modern bundle theoretic approach has increased greatly its understandability.

This book has a long history of more than 20 years. The first attempt to write a monograph on information-theoretic approach to thermodynamics was done by one of the authors (RSI) in 1974 when he published, in the preprint form, two volumes of the book "Information Theory and Thermodynamics" concerning classical and quantum information theory, [153] (220 pp.), [154] (185 pp.). In spite of the encouraging remarks by some of the readers, the physical part of this book was never written except for the first chapter. Now this material is written completely anew and in much greater extent. A few years earlier, in 1970, second author of the present book, (AK), a doctoral student and collaborator of RSI in Toruli, published in Polish, also as a preprint, his habilitation dissertation "Information-theoretical decision scheme in quantum statistical mechanics" [196] (96 pp.). This small monograph presented his original results in the physical part of the theory developed in the Torun school. Unfortunately, this preprint was never published in English. The present book contains all these results in a much more modern and developed form.

Roman Ingarden studied under Husserl before and during the first world war. He belonged to the so-called Gottingen group of Husserl''s pupils. Husserl''s doctrine was accepted by them and interpreted in a realist vein. Ingarden defended this view all his life. He opposed the development of phenomenology towards idealism. A considerable part of Ingarden''s great creative effort is dedicated to the construction of a realist phenomenology and thus, according to him, to continuing the erection of the theoret­ ical structure whose foundations were laid by Husserl in his Logical Investigations. From Ingarden''s standpoint the question of idealism versus realism was a crucial one. Ingarden published several studies on Husserl. The first one was written in 1918 and the last one was published posthumously. The present essay was printed in Ingarden''s book Z badan nad filozofi:t­ wsp61czesn:t- (Inquiries into Contemporary Philosophy 1963) along with a number of other essays on Husserl and his philoso­ phy. This one is representative for Ingarden''s positions. It is a good example of his contribution to an important controversy in the history of phenomenology, and it gives the reader an idea of Ingarden''s critique of Husserlian idealism against the background of his argument for realism. Thanks and acknowledgements are due to Mr. J. E. Llewelyn of Edinburgh University. This translation was undertaken in collaboration with him. Arn6r Hannibalsson K6pavogur, Iceland 2I. II.

The present book has been written by two mathematicians and one physicist: a pure mathematician specializing in Finsler geometry (Makoto Matsumoto), one working in mathematical biology (Peter Antonelli), and a mathematical physicist specializing in information thermodynamics (Roman Ingarden). The main purpose of this book is to present the principles and methods of sprays (path spaces) and Finsler spaces together with examples of applications to physical and life sciences. It is our aim to write an introductory book on Finsler geometry and its applications at a fairly advanced level. It is intended especially for graduate students in pure mathemat­ ics, science and applied mathematics, but should be also of interest to those pure "Finslerists" who would like to see their subject applied. After more than 70 years of relatively slow development Finsler geometry is now a modern subject with a large body of theorems and techniques and has math­ ematical content comparable to any field of modern differential geometry. The time has come to say this in full voice, against those who have thought Finsler geometry, because of its computational complexity, is only of marginal interest and with prac­ tically no interesting applications. Contrary to these outdated fossilized opinions, we believe "the world is Finslerian" in a true sense and we will try to show this in our application in thermodynamics, optics, ecology, evolution and developmental biology. On the other hand, while the complexity of the subject has not disappeared, the modern bundle theoretic approach has increased greatly its understandability.

The present book has been written by two mathematicians and one physicist: a pure mathematician specializing in Finsler geometry (Makoto Matsumoto), one working in mathematical biology (Peter Antonelli), and a mathematical physicist specializing in information thermodynamics (Roman Ingarden). The main purpose of this book is to present the principles and methods of sprays (path spaces) and Finsler spaces together with examples of applications to physical and life sciences. It is our aim to write an introductory book on Finsler geometry and its applications at a fairly advanced level. It is intended especially for graduate students in pure mathemat­ ics, science and applied mathematics, but should be also of interest to those pure "Finslerists" who would like to see their subject applied. After more than 70 years of relatively slow development Finsler geometry is now a modern subject with a large body of theorems and techniques and has math­ ematical content comparable to any field of modern differential geometry. The time has come to say this in full voice, against those who have thought Finsler geometry, because of its computational complexity, is only of marginal interest and with prac­ tically no interesting applications. Contrary to these outdated fossilized opinions, we believe "the world is Finslerian" in a true sense and we will try to show this in our application in thermodynamics, optics, ecology, evolution and developmental biology. On the other hand, while the complexity of the subject has not disappeared, the modern bundle theoretic approach has increased greatly its understandability.

This book has a long history of more than 20 years. The first attempt to write a monograph on information-theoretic approach to thermodynamics was done by one of the authors (RSI) in 1974 when he published, in the preprint form, two volumes of the book "Information Theory and Thermodynamics" concerning classical and quantum information theory, [153] (220 pp.), [154] (185 pp.). In spite of the encouraging remarks by some of the readers, the physical part of this book was never written except for the first chapter. Now this material is written completely anew and in much greater extent. A few years earlier, in 1970, second author of the present book, (AK), a doctoral student and collaborator of RSI in Toruli, published in Polish, also as a preprint, his habilitation dissertation "Information-theoretical decision scheme in quantum statistical mechanics" [196] (96 pp.). This small monograph presented his original results in the physical part of the theory developed in the Torun school. Unfortunately, this preprint was never published in English. The present book contains all these results in a much more modern and developed form.

Roman Ingarden studied under Husserl before and during the first world war. He belonged to the so-called Gottingen group of Husserl's pupils. Husserl's doctrine was accepted by them and interpreted in a realist vein. Ingarden defended this view all his life. He opposed the development of phenomenology towards idealism. A considerable part of Ingarden's great creative effort is dedicated to the construction of a realist phenomenology and thus, according to him, to continuing the erection of the theoret­ ical structure whose foundations were laid by Husserl in his Logical Investigations. From Ingarden's standpoint the question of idealism versus realism was a crucial one. Ingarden published several studies on Husserl. The first one was written in 1918 and the last one was published posthumously. The present essay was printed in Ingarden's book Z badan nad filozofi:t­ wsp61czesn:t- (Inquiries into Contemporary Philosophy 1963) along with a number of other essays on Husserl and his philoso­ phy. This one is representative for Ingarden's positions. It is a good example of his contribution to an important controversy in the history of phenomenology, and it gives the reader an idea of Ingarden's critique of Husserlian idealism against the background of his argument for realism. Thanks and acknowledgements are due to Mr. J. E. Llewelyn of Edinburgh University. This translation was undertaken in collaboration with him. Arn6r Hannibalsson K6pavogur, Iceland 2I. II.

The present book has been written by two mathematicians and one physicist: a pure mathematician specializing in Finsler geometry (Makoto Matsumoto), one working in mathematical biology (Peter Antonelli), and a mathematical physicist specializing in information thermodynamics (Roman Ingarden). The main purpose of this book is to present the principles and methods of sprays (path spaces) and Finsler spaces together with examples of applications to physical and life sciences. It is our aim to write an introductory book on Finsler geometry and its applications at a fairly advanced level. It is intended especially for graduate students in pure mathemat­ ics, science and applied mathematics, but should be also of interest to those pure "Finslerists" who would like to see their subject applied. After more than 70 years of relatively slow development Finsler geometry is now a modern subject with a large body of theorems and techniques and has math­ ematical content comparable to any field of modern differential geometry. The time has come to say this in full voice, against those who have thought Finsler geometry, because of its computational complexity, is only of marginal interest and with prac­ tically no interesting applications. Contrary to these outdated fossilized opinions, we believe "the world is Finslerian" in a true sense and we will try to show this in our application in thermodynamics, optics, ecology, evolution and developmental biology. On the other hand, while the complexity of the subject has not disappeared, the modern bundle theoretic approach has increased greatly its understandability.

This book has a long history of more than 20 years. The first attempt to write a monograph on information-theoretic approach to thermodynamics was done by one of the authors (RSI) in 1974 when he published, in the preprint form, two volumes of the book "Information Theory and Thermodynamics" concerning classical and quantum information theory, [153] (220 pp.), [154] (185 pp.). In spite of the encouraging remarks by some of the readers, the physical part of this book was never written except for the first chapter. Now this material is written completely anew and in much greater extent. A few years earlier, in 1970, second author of the present book, (AK), a doctoral student and collaborator of RSI in Toruli, published in Polish, also as a preprint, his habilitation dissertation "Information-theoretical decision scheme in quantum statistical mechanics" [196] (96 pp.). This small monograph presented his original results in the physical part of the theory developed in the Torun school. Unfortunately, this preprint was never published in English. The present book contains all these results in a much more modern and developed form.

Roman Ingarden studied under Husserl before and during the first world war. He belonged to the so-called Gottingen group of Husserl''s pupils. Husserl''s doctrine was accepted by them and interpreted in a realist vein. Ingarden defended this view all his life. He opposed the development of phenomenology towards idealism. A considerable part of Ingarden''s great creative effort is dedicated to the construction of a realist phenomenology and thus, according to him, to continuing the erection of the theoret­ ical structure whose foundations were laid by Husserl in his Logical Investigations. From Ingarden''s standpoint the question of idealism versus realism was a crucial one. Ingarden published several studies on Husserl. The first one was written in 1918 and the last one was published posthumously. The present essay was printed in Ingarden''s book Z badan nad filozofi:t­ wsp61czesn:t- (Inquiries into Contemporary Philosophy 1963) along with a number of other essays on Husserl and his philoso­ phy. This one is representative for Ingarden''s positions. It is a good example of his contribution to an important controversy in the history of phenomenology, and it gives the reader an idea of Ingarden''s critique of Husserlian idealism against the background of his argument for realism. Thanks and acknowledgements are due to Mr. J. E. Llewelyn of Edinburgh University. This translation was undertaken in collaboration with him. Arn6r Hannibalsson K6pavogur, Iceland 2I. II.

The present book has been written by two mathematicians and one physicist: a pure mathematician specializing in Finsler geometry (Makoto Matsumoto), one working in mathematical biology (Peter Antonelli), and a mathematical physicist specializing in information thermodynamics (Roman Ingarden). The main purpose of this book is to present the principles and methods of sprays (path spaces) and Finsler spaces together with examples of applications to physical and life sciences. It is our aim to write an introductory book on Finsler geometry and its applications at a fairly advanced level. It is intended especially for graduate students in pure mathemat­ ics, science and applied mathematics, but should be also of interest to those pure "Finslerists" who would like to see their subject applied. After more than 70 years of relatively slow development Finsler geometry is now a modern subject with a large body of theorems and techniques and has math­ ematical content comparable to any field of modern differential geometry. The time has come to say this in full voice, against those who have thought Finsler geometry, because of its computational complexity, is only of marginal interest and with prac­ tically no interesting applications. Contrary to these outdated fossilized opinions, we believe "the world is Finslerian" in a true sense and we will try to show this in our application in thermodynamics, optics, ecology, evolution and developmental biology. On the other hand, while the complexity of the subject has not disappeared, the modern bundle theoretic approach has increased greatly its understandability.