Combining Texts

All the ideas for 'Thinking About Mathematics', 'The Sentiment of Rationality' and 'On the Heavens'

expand these ideas     |    start again     |     specify just one area for these texts


45 ideas

1. Philosophy / A. Wisdom / 1. Nature of Wisdom
It is wisdom to believe what you desire, because belief is needed to achieve it [James]
1. Philosophy / D. Nature of Philosophy / 1. Philosophy
All good philosophers start from a dumb conviction about which truths can be revealed [James]
1. Philosophy / E. Nature of Metaphysics / 3. Metaphysical Systems
A complete system is just a classification of the whole world's ingredients [James]
2. Reason / A. Nature of Reason / 5. Objectivity
A single explanation must have a single point of view [James]
2. Reason / A. Nature of Reason / 9. Limits of Reason
A very hungry man cannot choose between equidistant piles of food [Aristotle]
2. Reason / B. Laws of Thought / 6. Ockham's Razor
Our greatest pleasure is the economy of reducing chaotic facts to one single fact [James]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
Cauchy gave a formal definition of a converging sequence. [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
Categories are the best foundation for mathematics [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
6. Mathematics / C. Sources of Mathematics / 7. Formalism
Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
'Impredicative' definitions refer to the thing being described [Shapiro]
7. Existence / E. Categories / 2. Categorisation
Classification can only ever be for a particular purpose [James]
12. Knowledge Sources / C. Rationalism / 1. Rationalism
Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro]
14. Science / A. Basis of Science / 1. Observation
Scientific genius extracts more than other people from the same evidence [James]
14. Science / A. Basis of Science / 6. Falsification
Experimenters assume the theory is true, and stick to it as long as result don't disappoint [James]
14. Science / C. Induction / 3. Limits of Induction
We can't know if the laws of nature are stable, but we must postulate it or assume it [James]
14. Science / C. Induction / 6. Bayes's Theorem
Trying to assess probabilities by mere calculation is absurd and impossible [James]
14. Science / D. Explanation / 2. Types of Explanation / j. Explanations by reduction
We have a passion for knowing the parts of something, rather than the whole [James]
15. Nature of Minds / A. Nature of Mind / 1. Mind / b. Purpose of mind
The mind has evolved entirely for practical interests, seen in our reflex actions [James]
15. Nature of Minds / A. Nature of Mind / 7. Animal Minds
Dogs' curiosity only concerns what will happen next [James]
18. Thought / A. Modes of Thought / 5. Rationality / a. Rationality
How can the ground of rationality be itself rational? [James]
18. Thought / A. Modes of Thought / 5. Rationality / b. Human rationality
It seems that we feel rational when we detect no irrationality [James]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / d. Biological ethics
Evolution suggests prevailing or survival as a new criterion of right and wrong [James]
22. Metaethics / B. Value / 2. Values / b. Successful function
Each thing that has a function is for the sake of that function [Aristotle]
26. Natural Theory / A. Speculations on Nature / 2. Natural Purpose / a. Final purpose
An unworn sandal is in vain, but nothing in nature is in vain [Aristotle]
There has to be some goal, and not just movement to infinity [Aristotle]
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / f. Ancient elements
Aether moves in circles and is imperishable; the four elements perish, and move in straight lines [Aristotle, by Gill,ML]
An element is what bodies are analysed into, and won't itself divide into something else [Aristotle]
26. Natural Theory / C. Causation / 8. Particular Causation / d. Selecting the cause
Understanding by means of causes is useless if they are not reduced to a minimum number [James]
27. Natural Reality / A. Classical Physics / 1. Mechanics / a. Explaining movement
If the more you raise some earth the faster it moves, why does the whole earth not move? [Aristotle]
27. Natural Reality / C. Space / 1. Void
Void is a kind of place, so it can't explain place [Aristotle]
27. Natural Reality / E. Cosmology / 1. Cosmology
The earth must be round and of limited size, because moving north or south makes different stars visible [Aristotle]
The Earth must be spherical, because it casts a convex shadow on the moon [Aristotle]
27. Natural Reality / E. Cosmology / 3. The Beginning
Everyone agrees that the world had a beginning, but thinkers disagree over whether it will end [Aristotle]
27. Natural Reality / E. Cosmology / 10. Multiverse
It seems possible that there exists a limited number of other worlds apart from this one [Aristotle]
29. Religion / B. Monotheistic Religion / 4. Christianity / a. Christianity
Early Christianity says God recognises the neglected weak and tender impulses [James]