The Emergence of First-Order LogicWhat is the foundation of logic ?
For anybody schooled in modern logic, first-order logic can seem an entirely natural object of study, and its discovery inevitable. It is semantically complete; it is adequate to the axiomatization of all ordinary mathematics; and Lindström’s theorem shows that it is the maximal logic satisfying the compactness and Löwenheim-Skolem properties.
So it is not surprising that first-order logic has long been regarded as the “right” logic for investigations into the foundations of mathematics. It occupies the central place in modern textbooks of mathematical logic, with other systems relegated to the sidelines.
The history, however, is anything but straightforward, and is certainly not a matter of a sudden discovery by a single researcher. The emergence is bound up with technical discoveries, with differing conceptions of what constitutes logic, with different programs of mathematical research, and with philosophical and conceptual reflection. So if first-order logic is “natural”, it is natural only in retrospect. The story is intricate, and at points contested; the following entry can only provide an overview.
https://plato.stanford.edu/entries/logic-firstorder-emergence/Discussions of various aspects of the development are provided by Goldfarb 1979, Moore 1988, Eklund 1996, Brady 2000, Ferreirós 2001, Sieg 2009, Mancosu, Zach & Badesa 2010, Schiemer & Reck 2013, the notes to Hilbert [LFL], and the encyclopedic handbook Gabbay & Woods 2009......more
The Emergence of First-Order Logic
First published Sat Nov 17, 2018 https://plato.stanford.edu/entries/logic-firstorder-emergence/
Logic is logic, in any form. It is a process. Mathematics is one of the purest forms of logic.This about mathematical logic .
And Mathematical logic is not the pinnacle of logic . Just a form of logic .
en.wikipedia.org › wiki › Mathematical_logicMathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. ... Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory.
These areas share basic results on logic, particularly first-order logic, and definability.
Logic is logic, in any form. It is a process. Mathematics is one of the purest forms of logic.
en.wikipedia.org › wiki › Mathematical_logic
Yes, it does not rely on any form of assumptive truths.Mathematics is " One of " . Forms of logic
: a science of developing and representing logical principles by means of a formalized system consisting of primitive symbols, combinations of these symbols, axioms, and rules of inference
Logical progression :Define " Pure logic " .
" Philosophy is written in this grand book – I mean the Universe – which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.Galileo, a contemporary of Descartes, also blurred the distinction between mathematical and philosophical method. An excerpt from his essay ‘Il Saggiatore’ (1623), or The Assayer, is often cited for advancing a revolutionary mathematisation of physics:
In this quotation, it is philosophy that is written in the language of mathematics. It is no mere linguistic coincidence that Isaac Newton’s monumental development of calculus and modern physics was titled Philosophiæ Naturalis Principia Mathematica (1687), that is, Mathematical Principles of Natural Philosophy. The goal of philosophy is to understand the world and our place in it, and to determine the methods that are appropriate to that task. Physics, or natural philosophy, was part of that project, and Descartes, Galileo and Newton – and philosophers before and after – were keenly attentive to the role that mathematics had to play.
The only way to be able to determine a magnitutude is to symbolize it "The turning point occurs around rule 14. According to Descartes, philosophy is a matter of discovering general truths by finding properties that are shared by disparate objects, in order to understand the features that they have in common. This requires comparing the degrees to which the properties occur. A property that admits degrees is, by definition, a magnitude.
And, from the time of the ancient Greeks, mathematics was understood to be neither more nor less than the science of magnitudes. (It was taken to encompass both the study of discrete magnitudes, that is, things that can be counted, as well as the study of continuous magnitudes, which are things that can be represented as lengths.)
https://aeon.co/essays/does-philosophy-still-need-mathematics-and-vice-versaPhilosophy is therefore the study of things that can be represented in mathematical terms, and the philosophical method becomes virtually indistinguishable from the mathematical method.
It does not rely on any form of assumptive truths.
All elements of basic (pure) mathematics are defined and definable in symbolic language.
Definition of symbolic logic
Please define a physical object without appealing to irreducible complexity, and how does it control the mathematics of its behavior?To the last statement ; disagree
The physical objects are not controlled by mathematics . Mathematics is controlled by real physical objects .
river said: ↑
To the last statement ; disagree
The physical objects are not controlled by mathematics . Mathematics is controlled by real physical objects .
How?
An old Aussie expression Write4U, debating this with river, is akin to pushing shit up hill.Please define a physical object without appealing to irreducible complexity, and how does it control the mathematics of its behavior?
An old Aussie expression Write4U, debating this with river, is akin to pushing shit up hill.
In defining maths, I like the simple definition as being the language of physics, but a more encompassing definition is in https://en.wikipedia.org/wiki/Mathematics
and defined as
"Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions."
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Note the highlighted bit. "Ëstablishing Truth"
This is the reason why river in general, dismisses the importance of maths, as it also defines and supports science/physics and the scientific method, and at the same time refuting his completely ridiculous scenarios and fairy tales.
The "concept" of mathematics existed long before humans came on the scene.Think , what is the Root of mathematics . Where does the concept of mathematics come from ?
The "concept" of mathematics existed long before humans came on the scene.
The Universe functioned flawlessly 14 billion years ago as it does today, thanks to its mathematical underpinnings.
Human symbolic mathematics are a result of observation of natural dynamical phenomena and invented the symbolic language to describe these values and functional mechanisms, which allows us to practise the science of physics (physical values and potentials) to begin with.
Humans did not invent mathematics, humans invented symbolic languages to describe observed universal mathematics of self-ordering patterns from Planck scale and up.
p.s. Note how often bad human mathematics have been written from incorrect interpretation of natural phenomena. At no time did the Universe present a false picture, we just did not see it correctly, at that time.
The geometry and energy potential of this Universe is the essential truth of all things, regardless of the existence of humans and it is able to evolve all dense physical patterns which can be observed, experienced, and described by human mathematical symbolic language as "reality".
"Something" from "Nothing" is more logical than "Irreducible Complexity from Nothing"...
When did that "accounting" start and who started it and what symbols were used to describe the relative values of the "accounted" properties?Mathematics basic mathematics was based on accounting . Real physical objects .
When did that "accounting" start and who started it and what symbols were used to describe the relative values of the "accounted" properties?
So, you're going with the "Something from Nothing" proposition?Neither exist in the first place .
Sure, but the change is always in accordance with some universal mathematical imperative, and we can observe, measure, and mathematically symbolize these changes in "value", from one state (pattern) into another state (pattern).What is observed, experienced is always changing
Neither exist in the first place .
So, you're going with the "Something from Nothing" proposition?
And how did these original inventors of "accounting" keep accounts of everything in the Universe?I think it was in ancient Sumar . You had your own mark .
You appear to be going with confusion.↑
No I'm going with something from something .
Space/time/universe is either eternal/infinite, or had a beginning, which is accepted at the BB. The spacetime evolved from an unknown state at the BB. As Lawrence Krauss pushes [and which I find reasonably plausible] is that perhaps the quantum foam from whence the BB and spacetime evolved is the most realistic definition of nothing that is possible. I mean its pretty damn close to "nothing" that we generally define, and as such far more likely and logical then any ID creation event. The quantum foam is that which is eternal if we accept that definition of nothing.So, you're going with the "Something from Nothing" proposition?
Not bad.Sure, but the change is always in accordance with some universal mathematical imperative, and we can observe, measure, and mathematically symbolize these changes in "value", from one state (pattern) into another state (pattern).