56 ideas
| 23247 | The need to act produces consciousness, and practical reason is the root of all reason [Fichte] |
| Full Idea: Consciousness of the real world proceeds from the need to act, not the other way around. …Practical reason is the root of all reason. | |
| From: Johann Fichte (The Vocation of Man [1800], 3.I) | |
| A reaction: Strongly agree with the last part. In my conceptual scheme 'sensible' behaviour (e.g. of animals) precedes, in every way, rational behaviour. Sensible attitudes to quantity and magnitude precede mathematical logic. Minds exist for navigation. |
| 23232 | Sufficient reason makes the transition from the particular to the general [Fichte] |
| Full Idea: The principle of sufficient reason is the point of transition from the particular, which is itself, to the general, which is outside it. | |
| From: Johann Fichte (The Vocation of Man [1800], 1) | |
| A reaction: Not sure I understand this, but it seems worth passing on. Personally I would say that we have a knack of generalising, triggered when we spot patterns. |
| 17774 | Definitions make our intuitions mathematically useful [Mayberry] |
| Full Idea: Definition provides us with the means for converting our intuitions into mathematically usable concepts. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.405-1) |
| 17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
| Full Idea: When you have proved something you know not only that it is true, but why it must be true. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.405-2) | |
| A reaction: Note the word 'must'. Presumably both the grounding and the necessitation of the truth are revealed. |
| 17962 | The truth-maker principle is that every truth has a sufficient truth-maker [Forrest] |
| Full Idea: Item x is said to be a sufficient truth-maker for truth-bearer p just in case necessarily if x exists then p is true. ...Every truth has a sufficient truth-maker. Hence, I take it, the sum of all sufficient truth-makers is a universal truth-maker. | |
| From: Peter Forrest (General Facts,Phys Necessity, and Metaph of Time [2006], 1) | |
| A reaction: Note that it is not 'necessary', because something else might make p true instead. |
| 17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry] |
| Full Idea: Set theory cannot be an axiomatic theory, because the very notion of an axiomatic theory makes no sense without it. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.413-2) | |
| A reaction: This will come as a surprise to Penelope Maddy, who battles with ways to accept the set theory axioms as the foundation of mathematics. Mayberry says that the basic set theory required is much more simple and intuitive. |
| 17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
| Full Idea: We can give a semi-categorical axiomatisation of set-theory (all that remains undetermined is the size of the set of urelements and the length of the sequence of ordinals). The system is second-order in formalisation. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.413-2) | |
| A reaction: I gather this means the models may not be isomorphic to one another (because they differ in size), but can be shown to isomorphic to some third ingredient. I think. Mayberry says this shows there is no such thing as non-Cantorian set theory. |
| 17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry] |
| Full Idea: The (misnamed!) Axiom of Infinity expresses Cantor's fundamental assumption that the species of natural numbers is finite in size. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.414-2) |
| 17801 | The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry] |
| Full Idea: The idea of 'generating' sets is only a metaphor - the existence of the hierarchy is established without appealing to such dubious notions. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.414-2) | |
| A reaction: Presumably there can be a 'dependence' or 'determination' relation which does not involve actual generation. |
| 17803 | Limitation of size is part of the very conception of a set [Mayberry] |
| Full Idea: Our very notion of a set is that of an extensional plurality limited in size. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.415-2) |
| 17786 | The mainstream of modern logic sees it as a branch of mathematics [Mayberry] |
| Full Idea: In the mainstream tradition of modern logic, beginning with Boole, Peirce and Schröder, descending through Löwenheim and Skolem to reach maturity with Tarski and his school ...saw logic as a branch of mathematics. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.410-1) | |
| A reaction: [The lesser tradition, of Frege and Russell, says mathematics is a branch of logic]. Mayberry says the Fregean tradition 'has almost died out'. |
| 17788 | First-order logic only has its main theorems because it is so weak [Mayberry] |
| Full Idea: First-order logic is very weak, but therein lies its strength. Its principle tools (Compactness, Completeness, Löwenheim-Skolem Theorems) can be established only because it is too weak to axiomatize either arithmetic or analysis. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.411-2) | |
| A reaction: He adds the proviso that this is 'unless we are dealing with structures on whose size we have placed an explicit, finite bound' (p.412-1). |
| 17791 | Only second-order logic can capture mathematical structure up to isomorphism [Mayberry] |
| Full Idea: Second-order logic is a powerful tool of definition: by means of it alone we can capture mathematical structure up to isomorphism using simple axiom systems. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1) |
| 17787 | Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry] |
| Full Idea: The 'logica magna' [of the Fregean tradition] has quantifiers ranging over a fixed domain, namely everything there is. In the Boolean tradition the domains differ from interpretation to interpretation. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.410-2) | |
| A reaction: Modal logic displays both approaches, with different systems for global and local domains. |
| 17790 | No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry] |
| Full Idea: No logic which can axiomatize real analysis can have the Löwenheim-Skolem property. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1) |
| 17778 | Axiomatiation relies on isomorphic structures being essentially the same [Mayberry] |
| Full Idea: The central dogma of the axiomatic method is this: isomorphic structures are mathematically indistinguishable in their essential properties. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.406-2) | |
| A reaction: Hence it is not that we have to settle for the success of a system 'up to isomorphism', since that was the original aim. The structures must differ in their non-essential properties, or they would be the same system. |
| 17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry] |
| Full Idea: The purpose of a 'classificatory' axiomatic theory is to single out an otherwise disparate species of structures by fixing certain features of morphology. ...The aim is to single out common features. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.406-2) |
| 17780 | 'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry] |
| Full Idea: The purpose of what I am calling 'eliminatory' axiomatic theories is precisely to eliminate from mathematics those peculiar ideal and abstract objects that, on the traditional view, constitute its subject matter. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.407-1) | |
| A reaction: A very interesting idea. I have a natural antipathy to 'abstract objects', because they really mess up what could otherwise be a very tidy ontology. What he describes might be better called 'ignoring' axioms. The objects may 'exist', but who cares? |
| 17789 | No logic which can axiomatise arithmetic can be compact or complete [Mayberry] |
| Full Idea: No logic which can axiomatise arithmetic can be compact or complete. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1) | |
| A reaction: I take this to be because there are new truths in the transfinite level (as well as the problem of incompleteness). |
| 17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry] |
| Full Idea: We eliminate the real numbers by giving an axiomatic definition of the species of complete ordered fields. These axioms are categorical (mutually isomorphic), and thus are mathematically indistinguishable. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.408-2) | |
| A reaction: Hence my clever mathematical friend says that it is a terrible misunderstanding to think that mathematics is about numbers. Mayberry says the reals are one ordered field, but mathematics now studies all ordered fields together. |
| 17782 | Greek quantities were concrete, and ratio and proportion were their science [Mayberry] |
| Full Idea: Quantities for Greeks were concrete things - lines, surfaces, solids, times, weights. At the centre of their science of quantity was the beautiful theory of ratio and proportion (...in which the notion of number does not appear!). | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.407-2) | |
| A reaction: [He credits Eudoxus, and cites Book V of Euclid] |
| 17781 | Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry] |
| Full Idea: The abstract objects of modern mathematics, the real numbers, were invented by the mathematicians of the seventeenth century in order to simplify and to generalize the Greek science of quantity. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.407-2) |
| 17799 | Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry] |
| Full Idea: In Cantor's new vision, the infinite, the genuine infinite, does not disappear, but presents itself in the guise of the absolute, as manifested in the species of all sets or the species of all ordinal numbers. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.414-2) |
| 17797 | Cantor extended the finite (rather than 'taming the infinite') [Mayberry] |
| Full Idea: We may describe Cantor's achievement by saying, not that he tamed the infinite, but that he extended the finite. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.414-2) |
| 17775 | If proof and definition are central, then mathematics needs and possesses foundations [Mayberry] |
| Full Idea: If we grant, as surely we must, the central importance of proof and definition, then we must also grant that mathematics not only needs, but in fact has, foundations. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.405-1) |
| 17776 | The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry] |
| Full Idea: The ultimate principles upon which mathematics rests are those to which mathematicians appeal without proof; and the primitive concepts of mathematics ...themselves are grasped directly, if grasped at all, without the mediation of definition. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.405-1) | |
| A reaction: This begs the question of whether the 'grasping' is purely a priori, or whether it derives from experience. I defend the latter, and Jenkins puts the case well. |
| 17777 | Foundations need concepts, definition rules, premises, and proof rules [Mayberry] |
| Full Idea: An account of the foundations of mathematics must specify four things: the primitive concepts for use in definitions, the rules governing definitions, the ultimate premises of proofs, and rules allowing advance from premises to conclusions. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.405-2) |
| 17804 | Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry] |
| Full Idea: No axiomatic theory, formal or informal, of first or of higher order can logically play a foundational role in mathematics. ...It is obvious that you cannot use the axiomatic method to explain what the axiomatic method is. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.415-2) |
| 17792 | 1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry] |
| Full Idea: The sole theoretical interest of first-order Peano arithmetic derives from the fact that it is a first-order reduct of a categorical second-order theory. Its axioms can be proved incomplete only because the second-order theory is categorical. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1) |
| 17793 | It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry] |
| Full Idea: If we did not know that the second-order axioms characterise the natural numbers up to isomorphism, we should have no reason to suppose, a priori, that first-order Peano Arithmetic should be complete. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1) |
| 17794 | Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry] |
| Full Idea: The idea that set theory must simply be identified with first-order Zermelo-Fraenkel is surprisingly widespread. ...The first-order axiomatic theory of sets is clearly inadequate as a foundation of mathematics. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-2) | |
| A reaction: [He is agreeing with a quotation from Skolem]. |
| 17802 | We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry] |
| Full Idea: One does not have to translate 'ordinary' mathematics into the Zermelo-Fraenkel system: ordinary mathematics comes embodied in that system. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.415-1) | |
| A reaction: Mayberry seems to be a particular fan of set theory as spelling out the underlying facts of mathematics, though it has to be second-order. |
| 17805 | Set theory is not just another axiomatised part of mathematics [Mayberry] |
| Full Idea: The fons et origo of all confusion is the view that set theory is just another axiomatic theory and the universe of sets just another mathematical structure. ...The universe of sets ...is the world that all mathematical structures inhabit. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.416-1) |
| 23227 | Each object has a precise number of properties, each to a precise degree [Fichte] |
| Full Idea: Each object has a definite number of properties, no more, no less. …Each of these objects possesses each of these properties to a definite degree. | |
| From: Johann Fichte (The Vocation of Man [1800], 1) | |
| A reaction: Quine flatly disagrees with this. Fichte offers no grounds for his claim. On the whole I think of properties as psychologically abstracted by us from holistic objects, so there is plenty of room for error. The underlying powers are real. |
| 23228 | The principle of activity and generation is found in a self-moving basic force [Fichte] |
| Full Idea: The principle of activity, of generation and becoming in and for itself, is purely in that force itself and not in anything outside it…; the force is not driven or set in motion, it sets itself in motion. | |
| From: Johann Fichte (The Vocation of Man [1800], 1) | |
| A reaction: A good account of primitive powers, as self-motivating forces. I can't think what else could be fundamental to nature. This whole passage of Fichte expounds a powers ontology. |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| Full Idea: The abstractness of the old fashioned real numbers has been replaced by generality in the modern theory of complete ordered fields. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.408-2) | |
| A reaction: In philosophy, I'm increasingly thinking that we should talk much more of 'generality', and a great deal less about 'universals'. (By which I don't mean that redness is just the set of red things). |
| 23241 | I am myself, but not the external object; so I only sense myself, and not the object [Fichte] |
| Full Idea: I sense in myself, not in the object, for I am myself and not the object; therefore I sense only myself and my condition, and not the condition of the object. | |
| From: Johann Fichte (The Vocation of Man [1800], 2) | |
| A reaction: I'm not clear why anyone would have total confidence in internal experience and almost no confidence in experience of externals. In daily life I am equally confident about both. In philosophical mode I make equally cautious judgements about both. |
| 21969 | Awareness of reality comes from the free activity of consciousness [Fichte] |
| Full Idea: It is the necessary faith in our freedom of power, in our own real activity, and in the definite laws of human action, which lies at the root of all our consciousness of a reality external to ourselves. | |
| From: Johann Fichte (The Vocation of Man [1800], p.98), quoted by A.W. Moore - The Evolution of Modern Metaphysics 06.4 | |
| A reaction: I'd love to know what the 'laws of human action' are. Is it what Hume was trying to do? Moore says there is an 'element of self-creation' in Fichte's account of the source of reality. This is Descartes' dream argument biting back. |
| 21966 | Self-consciousness is the basis of knowledge, and knowing something is knowing myself [Fichte] |
| Full Idea: The immediate consciousness of myself is the condition of all other consciousness; and I know a thing only in so far as I know that I know it; no element can enter into the latter cognition which is not contained in the former. | |
| From: Johann Fichte (The Vocation of Man [1800], p.37), quoted by A.W. Moore - The Evolution of Modern Metaphysics 06.2 | |
| A reaction: This strikes me as false, and a lot of intellectual contortion would be needed to believe it. Is knowing this pen is in front of me a case of knowing that I have knowledge of this pen, or is it just knowledge of this pen? [cf Kant 1781:A129] |
| 21967 | There is nothing to say about anything which is outside my consciousness [Fichte] |
| Full Idea: Of any connection beyond the limits of my consciousness I cannot speak. ...I cannot proceed a hair's breadth beyond this consciousness, any more than I can spring out of myself. | |
| From: Johann Fichte (The Vocation of Man [1800], p.74), quoted by A.W. Moore - The Evolution of Modern Metaphysics 06.3 | |
| A reaction: I can't see that this is any different from the idealism of Berkeley, although they get there from different starting points. Idealist seem unable to even begin explaining consciousness. |
| 23231 | I immediately know myself, and anything beyond that is an inference [Fichte] |
| Full Idea: Immediately I know only of myself. What I am able to know beyond that I am only able to know through inference. | |
| From: Johann Fichte (The Vocation of Man [1800], 1) | |
| A reaction: A direct descendant of the Cartesian Cogito, I assume. Personally, if I bang my head on a beam I take the beam to be a full paid-up member of reality. Is it not possible that he also knows himself through inference? Do animals infer reality? |
| 23246 | Faith is not knowledge; it is a decision of the will [Fichte] |
| Full Idea: Faith is no knowledge, but a decision of the will to recognise the validity of knowledge. | |
| From: Johann Fichte (The Vocation of Man [1800], 3.I) | |
| A reaction: What matters is the grounds for the decision. Mad conspiracy theories are decisions of the will which are false. Legitimate faith is an intuition of coherence which cannot be fully articulated. |
| 23245 | Knowledge can't be its own foundation; there has to be regress of higher and higher authorities [Fichte] |
| Full Idea: No knowledge can be its own foundation and proof. Every knowledge presupposes something still higher as its foundation, and this ascent has no end. | |
| From: Johann Fichte (The Vocation of Man [1800], 3.I) | |
| A reaction: A metaphor that's hard to visualise! He must have in mind a priori as well as empirical knowledge. The 'higher' levels don't seem to be God, but some region of absolute rationality, to which free minds have access. I think. |
| 23242 | Consciousness has two parts, passively receiving sensation, and actively causing productions [Fichte] |
| Full Idea: My immediate consciousness is composed of two constituent parts, the consciousness of my passivity, the sensation; and the consciousness of my activity, in the production of an object according to the principle of causality. | |
| From: Johann Fichte (The Vocation of Man [1800], 2) | |
| A reaction: Kind of obvious, but unusual to make this sharp binary division. Modern neuroscience strongly militates against any and every simple binary division of brain activities. |
| 23240 | We can't know by sight or hearing without realising that we are doing so [Fichte] |
| Full Idea: Q. Could you not perhaps know an object through sight or hearing without knowing that you are seeing or hearing? A. Not at all. | |
| From: Johann Fichte (The Vocation of Man [1800], 2) | |
| A reaction: A nice statement of the traditional view which seemed to be demolished by the discovery of blindsight. In the light of modern brain research, the views of the mind found in past philosophers mostly seem very naïve. |
| 23243 | Consciousness of external things is always accompanied by an unnoticed consciousness of self [Fichte] |
| Full Idea: Q. So that constantly and under all circumstances my consciousness of things outside of me is accompanied by an unnoticed consciousness of myself? A. Quite so. | |
| From: Johann Fichte (The Vocation of Man [1800], 2) | |
| A reaction: He should be more cautious about asserting the existence of something 'unnoticed'. The Earth's core is unnoticed by me, but there is plenty of evidence for it. Not so sure about unnoticed self. Still, I think central control of the mind is indispensable. |
| 23244 | Forming purposes is absolutely free, and produces something from nothing [Fichte] |
| Full Idea: My thinking and originating of a purpose is in its nature absolutely free and brings forth something from nothing. | |
| From: Johann Fichte (The Vocation of Man [1800], 3.I) | |
| A reaction: Modern fans of free will are more equivocal in their assertions, and would be uncomfortable bluntly claiming to 'get something from nothing'. But that's what free will is! Embrace it, or run for your life. |
| 23237 | The capacity for freedom is above the laws of nature, with its own power of purpose and will [Fichte] |
| Full Idea: This capacity [for freedom], once it exists, is in the servitude of a power which is higher than nature and quite free of its laws, the power of purposes, and the will. | |
| From: Johann Fichte (The Vocation of Man [1800], 1) | |
| A reaction: You would think this could only refer to God, but he in fact is referring to the power of human free will. The clearest statement I have found of the weird human exceptionalism implied by a strong commitment to free will. |
| 23235 | I want independent control of the fundamental cause of my decisions [Fichte] |
| Full Idea: I want to be independent - not to be in and through another but to be something for myself: and as such I want myself to be the fundamental cause of all my determinations. | |
| From: Johann Fichte (The Vocation of Man [1800], 1) | |
| A reaction: I think this sums up the absurdity of the concept of free will. The only reason he gives for his passionate belief in free will is that he desperately wants some imagined 'fundamental cause' for his action, and he wants full control of that chimera. |
| 23230 | Nature contains a fundamental force of thought [Fichte] |
| Full Idea: There is an original force of thought in nature just as there is an original formative force. | |
| From: Johann Fichte (The Vocation of Man [1800], 1) | |
| A reaction: I think this idea is false, but it helps to understand Fichte. |
| 23233 | The will is awareness of one of our inner natural forces [Fichte] |
| Full Idea: To will is to be immediately conscious of the activity of one of our inner natural forces. | |
| From: Johann Fichte (The Vocation of Man [1800], 1) | |
| A reaction: A more Nietzschean view would be that to will is to be conscious of the victor among our inner natural drives. It can't just be awareness of one force, because the will feels conflicts. |
| 23234 | I cannot change the nature which has been determined for me [Fichte] |
| Full Idea: I cannot will the intention of making myself something other than what I am determined to be by nature, for I don't make myself at all but nature makes me and whatever I become. | |
| From: Johann Fichte (The Vocation of Man [1800], 1) | |
| A reaction: I take this to be a lot more accurate than Sartre's claim that we can re-make ourselves, but Fichte doesn't seem quite right. Don't I get any credit at all if I give up smoking, or train myself to treat someone more sympathetically? |
| 23239 | The self is, apart from outward behaviour, a drive in your nature [Fichte] |
| Full Idea: This 'you' for which you show such a lively interest is, so far as it is not overt behaviour, at least a drive in your own peculiar nature. | |
| From: Johann Fichte (The Vocation of Man [1800], 1) | |
| A reaction: I assume this use of 'drive' is the origin of Nietzsche's picture of such things, focused on the basic will to power. I like Fichte's emphasis on active forces as the basis of nature. |
| 23238 | If life lacks love it becomes destruction [Fichte] |
| Full Idea: Only in love is there life; without it there is death and annihilation. | |
| From: Johann Fichte (The Vocation of Man [1800], 1) | |
| A reaction: He gives not context of justification for this sudden claim. Watching from a melancholy distance the current 2022 Russian invasion of Ukraine, I take this idea to be a profound truth. If you let go of love, you float away down a dark stream. |
| 23236 | Freedom means making yourself become true to your essential nature [Fichte] |
| Full Idea: I want to be free means: I myself want to make myself be whatever I will be. I would therefore …already have to be, in a certain sense, what I am to become, so that I could make myself be it. | |
| From: Johann Fichte (The Vocation of Man [1800], 1) | |
| A reaction: This is much closer to the existenial picture of the malleable self, which Fichte arrives out once he commits to his desperate desire to have free will. [Not sure if my gist captures what he says]. |
| 23229 | Nature is wholly interconnected, and the tiniest change affects everything [Fichte] |
| Full Idea: Nature is an interconnected whole; …you could shift no grain of sand from its spot without thereby, perhaps invisibly to your eyes, changing something in all parts of the immeasurable whole. | |
| From: Johann Fichte (The Vocation of Man [1800], 1) | |
| A reaction: Sounds like idealist daydreaming, but might it actually be true with respect to gravity? |