Mathematical Generalities: ‘Number’ may be termed as a general term, but real numbers, a sub-set of numbers, is sub-general. Clearly, it is a quality: “having one member, having two members, etc.”; and here one, two, etc., when taken as nominatives, lose their significance, and are based primarily only on the adjectival use. Hence the justification for the adjectival (qualitative) primacy of numbers as universals. While defining one kind of ‘general’ another sort of ‘general’ may naturally be involved in the definition, insofar as they pertain to an existent process and not when otherwise.
Why are numbers and shapes so exact? ‘One’, ‘two’, ‘point’, ‘line’, etc. are all exact. The operations on these notions are also intended to be exact. But irrational numbers are not so exact in measurement. If notions like ‘one’, ‘two’, ‘point’, ‘line’, etc. are defined to be so exact, then it is not by virtue of the exactness of these substantive notions, but instead, due to their being defined as exact. Their adjectival natures: ‘being a unity’, ‘being two unities’, ‘being a non-extended shape’, etc., are not so exact.
A quality cannot be exact, but may be defined to be exact. It is in terms of the exactness attributed to these notions by definition that the adjectives ‘one’, ‘two’, ‘point’, ‘line’, etc. are exact. This is why the impossibility of fixing these (and other) substantive notions as exact miss our attention. If in fact these are inexact, then there is justification for the inexactness of irrational, transcendental, and other numbers too.
If numbers and shapes are in fact inexact, then not only irrational numbers, transcendental numbers, etc., but all exact numbers and the mathematical structures should remain inexact if they have not been defined as exact. And if behind the exact definitions of exact numbers there are no exact universals, i.e., quantitative qualities? If the formation of numbers is by reference to experience (i.e., not from the absolute vacuum of non-experience), their formation is with respect to the quantitatively qualitative and thus inexact ontological universals of oneness, two-ness, point, line, etc.
Thus, mathematical structures, in all their detail, are a species of qualities, namely, quantitative qualities, defined to be exact and not naturally exact. Quantitative qualities are ontological universals, with their own connotative and denotative versions.
Natural numbers, therefore, are the origin of primitive mathematical experience, although complex numbers may be more general than all others in a purely mathematical manner of definition.
Bibliography
(1) Gravitational Coalescence Paradox and Cosmogenetic Causality in Quantum Astrophysical Cosmology, 647 pp., Berlin, 2018.
(2) Physics without Metaphysics? Categories of Second Generation Scientific Ontology, 386 pp., Frankfurt, 2015.
(3) Causal Ubiquity in Quantum Physics: A Superluminal and Local-Causal Physical Ontology, 361 pp., Frankfurt, 2014.
(4) Essential Cosmology and Philosophy for All: Gravitational Coalescence Cosmology, 92 pp., KDP Amazon, 2022, 2nd Edition.
(5) Essenzielle Kosmologie und Philosophie für alle: Gravitational-Koaleszenz-Kosmologie, 104 pp., KDP Amazon, 2022, 1st Edition.
Howdy Raphael Neelamkavil,
"Quantitative qualities are ontological universals, with their own connotative and denotative versions." These are just words in a different discipline, essentially incomprehensible outside it, like "open set." Your example of irrationals may be supplemented with transcendentals and open sets and uncountable infinities, none of which are considered "exact" in the discipline that treats them, just defined or derived rigorously.
"One" pit, "a" pit, "the" pit as perceived by a cat leaping over it have a common physical state, ontologically, which is not determined by the cat, or by human definitions stuck on it for communication. Within mathematics, pure and practical, your concerns have no force unless someone is "doing mathematics" wrongly. Then it's called error and calls for no great effort to stamp it out: just let it be, since it will be corrected or replaced by another excess soon enough.
Inside pure mathematics all is well, and existents do not exist to be examined for "exactness," substantially or adjectivally. It is a fabulous fantasy of great precision but no actuality that could be an existent. The cat's world has "the" pit over which it jumps, yes, but that the cat's perception introduces anything as a quantitative quality in a pre-mathematical philosophy leaving the pit not naturally exact doesn't ring true for me. (Yes, the cat is on a related thread.)
No end of common speech errors exist and all disciplines have excessively enthusiastic devotees - it's religious devotion to them - but the discipline is not at fault. It can be guided, and I have supported your efforts to that end including my reference to Pasteur and Lister changing how medicine was done, but it's not about words, it's about people having such diversity.
I am not convinced by this post. I do not think that the fuss here actually clarifies anything outside wording for insiders. What is your objective in your reader's lives?
Happy Trails, Len
Leonard Hall, I am not trying to correct or guide anyone beyond practicing mathematics. As you know, there is a general notion that mathematics is exact. I tried here to find out why it is not exact. I gave just one example. Not that the other sorts of numbers are not to be mentioned. I mentioned just one kind as an example.
Now, to say that everything is well in math is not my manner of thinking. If that were the case, nothing would have happened in any science. I just shared a little reflection.
Howdy Raphael Neelamkavil,
You have changed the question without indicating what the changes are. Do you want to leave our previous answers hanging out to dry that way? I think it would be better for readers to introduce your changes as a reply with emphasis on your revised ideas as they develop. It would be good information for young folks learning to think and learning to allow their thoughts to be enriched over time.
Courtesy is worth the bother, and you are generally quite courteous. Carefully thought out answers can become absurd when the question has been changed. Changing the question is an affliction on ResearchGate that need not be here, although I have seen it several times.
Happy Trails, Len
Leonard Hall, in fact the text does already indicate that I do not hold that the basic notions of math are exact. This is why I clarified the title a little. I never imagined that this would be considered as a change!
In fact, I want to revise the lead text too. I shall do it after a few days and then also inform the readers about it. I have pleasure to say this!
Is the tendency to reification of our concepts about processes outside of us a matter of conceptualization or pre-conceptual?
If the tendency is pre-conceptual, how to theorize of it?
S. Korner in his book "The Philosophy of Mathematics" discusses this very topic. I too was once bothered by the issue of the exactness, but not really exact, numbers, and Korner's essay is the best I read on the issue. It's been a long time since I read his book, but if I remember it correctly he boiled it down to the problem of applied mathematics--why is mathematics so useful? Putting it in the current context: Why does the exactness of definitions in mathematics lead to conclusions about the (inexact) sensations we have of the world? Contrast this with the Copenhagen Interpretation of Quantum Mechanics which in a sense divorces the mathematics from the "real" world we cannot observe, and insists that it applies only to the world we observe.
Alun Wyn-Jones, this is wonderful. I can experience the relevance of the contrast you brought in. Congrats!
I add to it Hugh Everett III's conclusion while applying QM to cosmology, where he says that it is better to accept it because the mathematics says it. I have not cited the exact words.
Here he exhibits a sort of blind adoration to math. My attempt nowadays has also been to find out where in mathematical applications the math could be believed as sufficiently or absolutely the best guidepost, and where it must be limited. But then arises the question whether we can suggest some areas where the math should not be trusted!
The bafflement continues....
I have been revising this short discussion paper of mine in RG. It is an attempt to correct some basic attitudes in physics. Just now I have written an introduction to it. Please read it here. In a few days I shall upload the whole lead-text of this discussion for your reading and comments. Here please find only the introduction:
FOUNDATIONS OF AXIOMATIC PHILOSOPHY AND SCIENCE
1. INTRODUCTION
I get surprised each time when some physicists tell me that either the electromagnetic (EM) or the gravitational (G) or both the forms of energy do not exist, but are to be treated or expressed as waves or particles propagated from material objects that of course exist. Some of them put in all their energies to show that both EM and G are mere mathematical fields, and not physically existent fields of energy propagations from bodies.
This is similar in effect to Newton and his followers thinking honestly and religiously that gravitation and other energies are just miraculously non-bodily actions at a distance without any propagation particles / wavicles.
Even in the 21stcentury, we must be sharply aware that from the past more than 120 years the General Theory of Relativity and its various versions have succeeded in casting and maintaining the power of a terrifying veil of mathematical miracles on the minds of many scientists – miracles such as the mere spacetime curvature being the meaning of gravitation and all other sorts of fields.
A similar veil has been installed on the minds of many physicists by quantum physics too. We do not discuss it here. Hence, I have constructed in four published books a systemic manner of understanding these problems. I do not claim perfection in any of my attempts. Hence, I keep perfecting my efforts in the course of years. The following is a very short attempt to summarize in this effort one important point in physics and in the philosophy of physics.
Questioning the Foundations of Physical Constants, Properties, and Qualities
https://www.researchgate.net/post/Questioning_the_Foundations_of_Physical_Constants_Properties_and_Qualities
How to Ground Science and Philosophy Together Axiomatically?
https://www.researchgate.net/post/How_to_Ground_Science_and_Philosophy_Together_Axiomatically
The Fallacies of Space, Time, and Spacetime in Physics
https://www.researchgate.net/post/The_Fallacies_of_Space_Time_and_Spacetime_in_Physics
Physical and Exact Sciences and Axiomatic Philosophy: Introducing Grounding (long text)
https://www.researchgate.net/post/Physical_and_Exact_Sciences_and_Axiomatic_Philosophy_Introducing_Grounding_long_text
Causality and Statistics: Their Levels of Effect and of Explanation
https://www.researchgate.net/post/Causality_and_Statistics_Their_Levels_of_Effect_and_of_Explanation
Einstein-Podolsky-Rosen Paradox and Non-Locality: Is Einstein a Monist?
https://www.researchgate.net/post/Einstein-Podolsky-Rosen_Paradox_and_Non-Locality_Is_Einstein_a_Monist
Spacetime Curvatures, Gravitational Waves, Gravitons, and Anti-Gravitons: Do They All Exist?
https://www.researchgate.net/post/Spacetime_Curvatures_Gravitational_Waves_Gravitons_and_Anti-Gravitons_Do_They_All_Exist
The Fate of “Source-Independence” in Electromagnetism, Gravitation, and Monopoles
https://www.researchgate.net/post/The_Fate_of_Source-Independence_in_Electromagnetism_Gravitation_and_Monopoles
If anyone wants to read about a fantastic concept of the existence of consciousness and non-existence of the cosmos, see the comments till today here, by L Kurt Engelhart.......!!! Here you can learn the meaning of solipsism.... and similar to mathematical platonism creating ideas / notions into objects.
https://www.researchgate.net/post/How_to_philosophize_How_to_philosophize_in_the_sciences
Can what are termed "mathematically consistent" natural laws necessarily be physically consistent? If they can, then mathematics becomes physics (and for that matter any other mathematical science). But if math is different from all these sciences, the adequacy and applicability of math to physics and other sciences cannot be 100%. If that is the case, it is very important that physics (and other sciences) be helped to choose the most suitable math. This help can come from the same science/s only in a partially realized manner. Nor can math take up this task fully well. Hence, a generic science beyond all these including math and logic must take charge of the remaining portions of inadequacy and inapplicability of math to physics and the sciences. Which could that science be? I hold that this most general science need not contain all that philosophy has so far understood itself to be. But something of the philosophy of these sciences combined with the philosophy of math, logic, etc. would be an ideal option.
Essential Reason in Physicists’ Use of Logic: And in Other Sciences Too!
https://www.researchgate.net/post/Essential_Reason_in_Physicists_Use_of_Logic_And_in_Other_Sciences_Too
Preprint ESSENTIAL REASON IN PHYSICISTS' USE OF LOGIC: IN OTHER SCIENCES TOO
How Does Physics Know? The Epistemology Presupposed by Physics and Other Sciences
https://www.researchgate.net/post/How_Does_Physics_Know_The_Epistemology_Presupposed_by_Physics_and_Other_Sciences
Preprint MATHEMATICAL SOURCE OF FLAWS IN COSMOLOGICAL THEORIES: MATHE...
Preprint THE EPISTEMOLOGY PRESUPPOSED BY PHYSICS AND OTHER SCIENCES R...
PHYSICAL-PROCESSUAL REPRESENTATION OF IRRATIONAL NUMBERS
https://www.researchgate.net/post/Physical-Processual_Representation_of_Irrational_Numbers
THE ONTOLOGY BEHIND PHYSICS
3.1. Traditional Physical Categories
https://www.researchgate.net/post/The_Ontology_behind_Physics
https://www.researchgate.net/post/Grounded_Physical-Ontological_Categories_behind_Physics
Grounded (New) Physical-Ontological Categories behind Physics
Preprint THE ONTOLOGY BEHIND PHYSICS: CRITIQUE OF TRADITIONAL PHYSICA...
A SIMPLE GAME-CHANGER CAUSALITY FOR PHYSICS: Beyond the Two Millennia
https://www.researchgate.net/post/A_SIMPLE_GAME-CHANGER_CAUSALITY_FOR_PHYSICS_Beyond_the_Two_Millennia
https://www.researchgate.net/post/DEFINITION_OF_THE_ONTOLOGY_BEHIND_PHYSICS_5_Paragraphs
DEFINITION OF THE ONTOLOGY BEHIND PHYSICS (5 Paragraphs)
THE ANOMALY IN MATHEMATICAL / THEORETICAL PHYSICS (Short Text)
https://www.researchgate.net/post/THE_ANOMALY_IN_MATHEMATICAL_THEORETICAL_PHYSICS_Short_Text
Here a serious and somewhat complex matter to discuss:
NON-FOUNDATIONS OF ‘WAVICLES’ IN EINSTEIN-PODOLSKY-ROSEN PARADOX: Bases for Quantum Physics to Evolve (Maybe a physical-ontological Breakthrough)
https://www.researchgate.net/post/NON-FOUNDATIONS_OF_WAVICLES_IN_EINSTEIN-PODOLSKY-ROSEN_PARADOX_Bases_for_Quantum_Physics_to_Evolve_Maybe_a_physical-ontological_Breakthrough
AGAINST COSMIC ISOTROPY, CONFORMAL CYCLIC COSMOS, ETERNAL INFLATION, etc.: A Critique of Identity, Simultaneity, Cosmic Repetition / Recycling, etc.
https://www.researchgate.net/post/AGAINST_COSMIC_ISOTROPY_CONFORMAL_CYCLIC_COSMOS_ETERNAL_INFLATION_etc_A_Critique_of_Identity_Simultaneity_Cosmic_Repetition_Recycling_etc
Preprint A SIMPLE GAME-CHANGER CAUSALITY FOR PHYSICS Beyond the Two Millennia
Preprint ESSENTIAL LOGIC, EPISTEMOLOGY, AND ONTOLOGY BEHIND PHYSICS, ...
WHAT IS THE MYSTERIOUS STUFF OF INFORMATION? A Short but Clear Definition
https://www.researchgate.net/post/WHAT_IS_THE_MYSTERIOUS_STUFF_OF_INFORMATION_A_Short_but_Clear_Definition
Preprint COSMIC ISOTROPY, CONFORMAL CYCLIC COSMOS, ETERNAL INFLATION:...
THE PLANCK ERA / QUANTUM ERA and “DISAPPEARANCE” OF PHYSICAL CAUSALITY: “OMNIPOTENCE” OF MATHEMATICS
https://www.researchgate.net/post/THE_PLANCK_ERA_QUANTUM_ERA_and_DISAPPEARANCE_OF_PHYSICAL_CAUSALITY_OMNIPOTENCE_OF_MATHEMATICS
Preprint PLANCK ERA or QUANTUM ERA,and ”DISAPPEARANCE” OF CAUSALITY. ...
Preprint CAUSAL HORIZONAL RESEARCH: A METHODOLOGY IN PHYSICS Raphael ...
https://www.researchgate.net/post/WHAT_IS_INFORMATION_WHAT_IS_ITS_CAUSAL_OR_NON-CAUSAL_CORE_A_Discussion
https://www.researchgate.net/post/LINGUISTIC_HERESY_BEHIND_SELF-ORGANIZATION_SELF-REFERENCE_INTENTIONALITY_PHYSICAL_AND_BIOLOGICAL_SELF-INTERACTION_etc
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Preprint LINGUISTIC HERESY OF DENOTATIVE ABSOLUTISM: PHYSICAL-BIOLOGI...
https://www.researchgate.net/post/ONTOLOGICAL_DIFFERENCES_OF_CHARACTERISTICS_OF_ARTIFICIAL_AND_BIOLOGICAL_INTELLIGENCE_ALGORITHMS_AND_PROCEDURES_Against_Exaggerations
https://www.researchgate.net/post/WHY_EXACTLY_THE_WAVE-PARTICLE_DUALITY_Phenomenal_Ontological_Commitment_POC_as_the_Solution
https://www.researchgate.net/post/UNTENABLE_REIFICATION_OF_CONCEPTS_IN_PHYSICS_With_Examples
Preprint WHY EXACTLY WAVE-PARTICLE DUALITY? Phenomenal Ontological Co...
https://www.researchgate.net/post/DO_PHYSICAL_QUANTA_EXIST_Why_Should_CONSCIOUSNESS_Be_Treated_Quantum-Biologically
https://www.researchgate.net/post/HEIDEGGER_How_a_Philosopher_Destroys_His_Own_Thoughts_Coherence_and_Adequacy
Preprint UNIVERSAL CAUSALITY AND THE PHYSICAL-ONTOLOGICAL DEFECT OF N...
Preprint DIFFERENCES IN THE CONCEPTS OF CAUSALITY IN METAPHYSICS AND ...
Preprint BEYOND CAUSAL ITERATION QUANTIFIABILITY IN LINGUISTIC SPACE-TIME
Preprint BEYOND THE CAUSAL ITERATION METHOD. Short Text (Beyond Judea Pearl)
Preprint REFERENCE, APPLICABILITY, AND ADEQUACY OF UNIVERSALS, INFORM...
Preprint DENOTATIVE ABSOLUTISM. A 20TH CENTURY LOGICAL AND LINGUISTIC HERESY
Preprint INEVITABILITY OF COSMOLOGICAL, ONTOLOGICAL, AND EPISTEMOLOGI...
Preprint Introducing GRAVITATIONAL COALESCENCE PARADOX: COSMOGENETIC CAUSALITY
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