The limits of mathematics

Ian,

Thank you for this forum. The linkage between mathematics and physics and its limits is the most important foundational question. My work developing www.zenophysics.com indicates that much of the trouble with physics can be traced to the failure of Newton's calculus to counter Zeno's paradox of motion. Here is the story:

1. Zeno's paradox of motion: If you assume that points are dimensionless and that space is a continuum then Zeno's logic is accurate and motion is "logically" impossible. Take a particle like an electron at a position in space. For it to move it must first move to the point closest to it. The point closest to the electron is at the position of the electron because as a point it is without extent (dimensionless). All the points that the electron must move through are similarly at the position of the electron. There is no motion possible for the electron because the sequence of points that it must proceed through are at the position of the electron, and thus the electron cannot move. The continuum of points is quicksand for motion. QED

Of course something is wrong with the motion paradox. But is the problem with the Mathematical assumptions or the Physical assumptions in the paradox?

2. Can mathematics be changed to remedy this "loophole" that Zeno found? Alfred North Whitehead tried with a "point free" geometry. This concept has not picked up critical mass and the non dimensional point is still the preferred concept. Because of the tenacious of our concept of the point and the continuum I will make them the axiomatic start of physics and say that the start of physics is objects existing at point positions on the continuum of space-time. Math as it stands is basically OK.

3. If the math is OK then something is wrong with the physics in that it allows Zeno's trickery. This trouble was first spotted and remedied by Werner Heisenberg with his discrete quantum jumps which were part of his matrix mechanics formulation for quantum mechanics. Now the electron never has to traverse the points closest to it, it simply jumps over them. Heisenberg overcame Zeno, but he did not overcome Schroedinger and his continuous wave equation representation for quantum mechanics. Heisenberg had it right, reality is that physics has as its start discontinuous phenomena (see zenophysics). Trying to model this phenomena with continuous models will always be flawed.

4. The notion of discrete quantum jumps was put on the shelf. It needs to be taken off the shelf and dusted off and put into service to remedy the trouble with physics. See www.zenophysics.com for how I approach this. I need to proceed from Zeno to the other great enlightened trouble maker Isaac Newton.

5. Newton (and Galileo) flew in the face of Zeno by measuring motion. Motion is real because we can measure it as a velocity that is defined as distance divided by time v=deltaX/deltaT. So far so good.

6. Newton now performs the operation that causes the trouble with physics, he says that by using "logic" we can have velocity at a point and stick it to Zeno. He does this by showing that v=deltaX/deltaT=dx/dt. Every calculus text has a chapter where it is shown that deltaX can be taken to dx in the limit and deltaT can be taken to dt in the limit to produce the concept that a velocity can exist at a point. If the people who look at this chapter are mathematicians they are treated to a valuable exercise in logic. If the people who look at this chapter are physicists they are getting a dose of flapdoodle, because the real world does not work like that.

7. It looks like Newton showed that there is a physical reality to the notion of velocity at a point. He did no such thing. What he did do was split mathematics into two realms the unreal theoretical and the practical. It is in the unreal theoretical math world he defeats Zeno in a realm with no reality. In the world of real physics v=deltaX/deltaT is just fine as long as the deltaX is greater than the wavelength of the object under consideration.

8. An electron at a position in space has no velocity. However, you can calculate a velocity for the electron by measuring a deltaX and a deltaT and making a calculation for velocity.

The methods of calculus have to be rethought and brought up to date in light of the findings of quantum mechanics. This includes the notions of Hamiltonians and Hilbert spaces. Don L. 7/5/2010

Dear Don Limuti,

I also consider the basis of mathematic most fundamental to physics.

If you agree with Euclid that a point has no parts and with Peirce that a each part of continuum has parts then a continuum cannot be resolved in a finite amount of single points and there is no closest point to any point. This is not yet a physical but a mathematical explanation.

What "trouble in physics" did you resolve?

Given I am correct then quantum computing will not work as promised, several paradoxes will vanish, the LHC will not fulfill expectations, and several so far mandatory tenets will turn out at least questionable.

I found out:

1) Pupils of Gauss changed the good old notion of number as a measure to the rather questionable older understanding as a pebble.

2) IR is equivalent to IR and fits better to virtually all original physical quantities with natural zero including volume, area, distance, elapsed time, radius, temperature, pressure, number of electrons and other items ... , probability, ...

3) According to 2) Analysis of measured data does not require a complex valued Fourier transformation but just a real-valued Cosine transformation. The FT is redundant. In addition to the information available with the CT it only provides information on the arbitrarily chosen point of reference.

4) The fathers of QM introduced unilateral functions of frequency directly in a complex domain in a manner they borrowed from common practice where unilateral functions of time are considered the original ones. However they changed to the Hamiltonian point of view. This led to misinterpretation.

5) Perhaps, length contraction does not likewise belong to decreasing and increasing distance.

6) Space-time seems to be ill-founded because it would be anticipatory, and the sign of rotation clockwise or anticlockwise is arbitrary.

7) G. Cantor and his proponents did not refute Galilei's argument: The relations smaller, equal to, and larger are not valid for infinite quantities. As Fraenkel admitted: There is a 4th logical possibility: incomparability, and Cantor's naive definition of a set is untenable.

Regards,

Eckard

Dear Eckard,

The trouble with physics: The axioms of geometry* do not allow quantum particles to move.

I propose to makes the axioms of geometry legitimate for quantum particles by making particles move in a special way. That way of moving is what I call wavelength-hopping. The details are on www.digitalwavetheory.com

Thanks,

Don L.

* 1. Dimensionless Points 2. A continuum of points on a line.

The origin of special relativity has much to do with the origin of dynamics.

Useful axioms or theorems that could act as fundaments for physics exist, but they treat only the static relational structure of physical items and physical fields. These law sets consist for a part of traditional quantum logic as it is defined by Birkhoff, von Neumann, Jauch, Piron and others. Quantum logic says nothing about dynamics and it says nothing about physical fields. The static configuration of fields is treated in the Helmholtz and Hodge theorems. Dynamics couple these static ingredients. When dynamics occurs in steps, then after each step the static status quo is reestablished with new conditions. The laws governing the static status quo are reinstalled as well. In this picture nature bumps in universe wide steps from one static condition to the next one. With other words there exists a global progression parameter. This is not our common notion of (coordinate) time, because that has only local validity. In the new condition the actual configuration of the fields determine what will happen with the influenced physical items in the next step. The physical items are sources (charges) of the fields and when they move they form the currents that cause still other kinds of fields. Investigation of inertia shows that a movement that differs from a uniform movement and a non-uniform distribution of charges or currents cause forces that control the next step. In this way dynamics emerges.

All physical items in universe cooperate in the implementation of inertia. Because electrical charges can compensate each other while masses cannot do that inertia shows up for gravitational fields and not for electromagnetic fields. However, both fields take part in the emergence of dynamics.

The axioms that define the static interrelations between physical items stem from quantum logic. The sketched picture implies that the lattice of propositions about physical systems is controlled by influences that act as a sticky resistance against altering the relational structure of the set of propositions. In nature this sticky resistance is implemented by physical fields. Locally the action of the fields alter the observed properties of the physical items. It is done such that the combination of space and local time gets a Minkowski structure. In this way we get two new notions of time: coordinate time and proper time.

The global progression parameter is not a dimension of a space with Minkowski signature. Thus with this parameter the fields do not encounter a maximum speed of information transfer.

The former statement means that fields are not living in a Minkowski space. How is it then possible that our observable world has a Minkowski signature? It is only possible when during the step a conversion takes place that reshapes the expectation value of the position observable into something that has a Minkowski signature. Something that is capable of this is an infinitesimal transform of a hyper complex number q by a hyper complex number u that is close to unity:

[math]\textit{u}\approx 1+ \Delta \textit{s}[/math]

The transform runs:

[math] \Delta \textit{q} = \textit{u}\cdot \textit{q}/\textit{u} - \textit{q} [/math]

The infinitesimal steps of s and q are imaginary and orthogonal. The rectangular triangle can be closed with an infinitesimal step in the form of the hypotenuse. The corresponding variable is the coordinate time t. Now holds:

[math] \Delta \textit{s}^{2} = \Delta \textit{t}^{2} - \Delta \textit{q}^{2} [/math]

This construct has a Minkowski signature. In order to achieve this, q and u must be quaternions or higher dimensional hyper complex numbers. During the process the real part of q is not changed. It is not participating in the q step. The empty place is filled by the coordinate time t.

If this story is true then the space variable q and the coordinate time t have little in common. They do not form a spacetime construct. However the dynamically changed imaginary part of q forms together with t a Minkowskian spacetime.

Einstein left us his formulas for special relativity, but he never gave an explanation WHY these formulas work. The isomorphism of the lattice of Hilbert subspaces with the lattice of quantum logical propositions suggests that properties of physical items are represented by eigenvalues or as expectation values of normal operators in Hilbert space. Spacetime, with its Minkowski signature does not fit as an eigenvalue of a normal operator. Above an explanation is given for the fact that hyper complex numbers that can act as eigenvalues are capable of converting hyper complex numbers into constructs that indeed have a Minkowski signature.

The transform is indirectly done by a thing called redefiner. In quantum logic it redefines quantum logical propositions and its equivalent in Hilbert space redefines the subspaces that represent both the proposition and the physical item that is described by that proposition. The redefiner does not touch the Hilbert vectors. Thus it does not touch eigenvectors of operators. It represents the action of the Schrödinger picture. In the Heisenberg picture a set of parallel trails of unitary operators cooperate in order to shift the Hilbert subspace that corresponds with the considered physical item. The unitary transforms also shift the eigenvectors of operators. Since Hilbert space acts as an affine space for the change of the representation of the physical item, in both pictures the expectation value of the observed property of the physical item must be the same. In each step the set of unitary transforms cause an infinitesimal transform of the expectation value of q in the form

[math] \Delta \textit{q} = \textit{u}\cdot \textit{q}/\textit{u} - \textit{q} [/math]

as sketched above.

More details can be found in http://www.scitech.nl/English/Science/Ontheoriginofdynamics.pdf.

I am new to this forum, so I forgot to add my name and I did not introduce myself. I am a retired physist. I spend my retirement on thinking what is good and what is bad about contempory physics. By browsing around I encountered a series of important inconsistencies. All have to to with the incorrect implementation of mathematics. An example is the fact that according to the isomorphism between the lattice of quantum logical propositions and the lattice of closed subspaces of an inifinite separatable Hilbert space the quantum propositions can be represented by the corresponding closed subspaces. If everything that can be said about a physical item is formulated as a proposition, then that proposition represents the physical item and the corresponding subspace also represents that item. Why then will most physicists represent physical items by wave functions or by density operators. The wave function corresponds with a single Hilbert vector instead of with a multidimensional subspace. How can a single Hilbert vector carry all the properties of the DNA molecule that represents on its turn the characteristics of Napoleon? A multdimensional subspace would fit far better.

There are not so much divergences between physics and mathematics, but instead there exist many inconsistencies in the way mathematics is applied to physics. This is not the failure of physics. It is the failure of the physicists.

Greathings, Hans

Dear All,

Mathematical representation of absolute truth, zero = i = infinity can be deduced as follows

If 0 x 0 = 0 is true, then 0 / 0 = 0 is also true

If 0 x 1 = 0 is true, then 0 / 0 = 1 is also true

If 0 x 2 = 0 is true, then 0 / 0 = 2 is also true

If 0 x i = 0 is true, then 0 / 0 = i is also true

If 0 x ~ = 0 is true, then 0 / 0 = ~ is also true

It seems that mathematics, the universal language, is also pointing to the absolute truth that 0 = 1 = 2 = i = ~, where "i" can be any number from zero to infinity. We have been looking at only first half of the if true statements in the relative world. As we can see it is not complete with out the then true statements whic are equally true. As all numbers are equal mathematically, so is all creation equal "absolutely".

This proves that 0 = i = ~ or in words "absolutely" nothing = "relatively" everything or everything is absolutely equal. Singularity is not only relative infinity but also absolute equality. There is only one singularity or infinity in the relativistic universe and there is only singularity or equality in the absolute universe and we are all in it.

Love,

Sridattadev.

This is an elementary slapdash answer before I lose my cool and run away from this conversation.

You state above: "What I am interested in from a non-speculative standpoint are those places that mathematics *diverges* from the physical world, i.e. those places in which mathematics describes something that is very clearly not physically possible. Now, this obviously does not mean any instance of infinity since, as an example, infinity is necessary to resolve such physical paradoxes as those conjectured by Zeno. But there *are* instances where mathematics in some infinite limit ends up describing something that is not physically possible. There are probably plenty of examples that don't involve infinity."

My intuition and mathematical researches lead me to feel strongly that insight into the above problem can be gained by a fuller understanding of the relationships between Thought, Mathematics, and Physical reality. This can be accomplished by comparing the math describing both thought and reality. The mathematics (symbology) is the same in each case, even though the two Subjects: Thought and Reality are different Kinds of subjects. So the math contains the clues.

Two specifics come to mind: 1)George Boole's law of thought: X^2=X, and 2)the Quantum Mechanical linear operator |Y>

Dang, I had a totally brilliant thread there for you to contemplate, and then deleted most of it with an accidental keystroke. But because of the lack of feedback I get in these threads, that should be no problem for anyone. So, I ain't worth retyping. Here I'll just finish the last cut off sentence above:

2) the quantum mechanical linear operator |Y>

WTF! Warning, do not attempt to type the linear operator |Y> .

[|Y>

The bra and kets have to be conjugate imaginaries of each other for this equation to hold, of course...

Dear Sir,

Number is a property of all substances by which we differentiate between similars. If there are no similars, it is one. If there are similars, the number is many. Depending upon the sequence of perception of "one's", many can be 2, 3, 4...n etc. Mathematics is accumulation and reduction of similars, i.e., numbers of the same class of objects (like atomic number or mass number), which describes the changes in the physical phenomena or object when the numbers of any of the parameters are changed.

Mathematics is related to the result of measurement. Measurement is a conscious process of comparison between two similar quantities, one of which is called the scaling constant (unit). The cognition part induces the action leading to comparison, the reaction of which is again cognized as information. There is a threshold limit for such cognition. Hence Nature is mathematical in some perceptible ways. This has been proved by the German physiologist Mr. Ernst Heinrich Weber, who measured human response to various physical stimuli. Carrying out experiments with lifting increasing weights, he devised the formula: ds = k (dW / W), where ds is the threshold increase in response (the smallest increase still discernible), dW the corresponding increase in weight, W the weight already present and k the proportionality constant. This has been developed as the Weber-Fechner law. This shows that the conscious response follows a somewhat logarithmic law. This has been successfully applied to a wide range of physiological responses.

Measurement is not the action of putting a scale to a rod, which is a mechanical action. Measurement is a conscious process of reaching an inference based on the action of comparison of something with an appropriate unit at "here-now". The readings of a particular aspect, which indicate a specific state of the object at a designated instant, (out of an infinite set of temporally evolving states), is frozen for use at other times and is known as the "result of measurement". The states relating to that aspect at all "other times", which cannot be measured; hence remain unknown, are clubbed together and are collectively referred to as the "superposition of states" (we call it adhyaasa). This concept has not only been misunderstood, but also unnecessarily glamorized and made incomprehensible in the "undead" Schrödinger's cat and other examples. The normal time evolution of the cat (its existential aspect) and the effect of its exposure to poisonous gas (the operational aspect) are two different unrelated aspects of its history. Yet these unrelated aspects have been coupled to bring in a state of coupled-superposition (we call it aadhyaasika taadaatmya), which is mathematically, physically and conceptually void.

Mathematics is related to accumulation and reduction of numbers. Since measurements are comparison between similar quantities, mathematics is possible only between similars (linear) or partly similars (non-linear) but never between the dissimilars. We cannot add or multiply 3 protons and 3 neutrons. They can be added only by taking their common property of mass to give mass number. These accumulation and reduction of numbers are expressed as the result of measurement after comparison with a scaling constant (standard unit) having similar characteristics (such as length compared with unit length, area with unit area, volume with unit volume, density with unit density, interval with unit interval, etc). The results of measurements are always pure numbers, i.e., scalar quantities, because the dimensions of the scaling constants are same for both the measuring device and the object being measured and measurement is only the operation of scaling up or down the unit for an appropriate number of times. Thus, mathematics explains only "how much" one quantity accumulates or reduces in an interaction involving similar or partly similar quantities and not "what", "why", "when", "where", or "with whom" about the objects involved in such interactions. These are the subject matters of physics. We will show repeatedly that in modern physics there is a mismatch and mix-up between the data, the mathematics and the physical theory.

Quantum physics implied that physical quantities usually have no values until they are observed. Therefore, the observer must be intrinsically involved in the physics being observed. This has been wrongly interpreted to mean that there might be no real world in the absence of an observer! When we measure a particular quantity, we come up with a specific value. This value is "known" only after the conscious or sentient content is added to the measurement. Thus, it is reasonable to believe that when we do not measure or perceive, we do not "know" the value - there is no operation of the conscious or sentient content is inert - and not that the quantity does not have any existential value. Here the failure of the physicists to find the correct "mathematics" to support their "theory" has been put forth as a pretext for denying reality. Mathematics is an expression of Nature, not its sole language. Though observer has a central role in Quantum theories, its true nature and mechanism has eluded the scientists. There cannot be an equation to describe the observer, the glory of the rising sun, the grandeur of the towering mountain, the numbing expanse of the night sky, the enchanting fragrance of the wild flower or the endearing smile on the lips of the beloved. It is not the same as any physical or chemical reaction or curvature of lips.

Mathematics is often manipulated to spread the cult of incomprehensibility. The electroweak theory is extremely speculative and uses questionable mathematics as a cover for opacity to predict an elusive Higg's mechanism. Yet, tens of millions of meaningless papers have been read out in millions of seminars world wide based on such unverified myth for a half century and more wasting enormous amounts of resources that could otherwise have been used to make the Earth a better place to live. The physicists use data from the excellent work done by experimental scientists to develop theories based on reverse calculation to match the result. It is nothing but politics of physics - claim credit for bringing in water in the river when it rains. Experiment without the backing of theory is blind. It can lead to disaster. Rain also brings floods. Experiments guided by economic and military considerations have brought havoc to our lives.

Regards,

basudeba

One real Universe is eternally occurring in one real here appearing moving in one real infinite dimension at one real universal "speed" of light, once. One real Universe can only be accurately describing by the abstract postulated symbol 1, once. Newton was wrong when he associated abstract planetary and astral motion with abstract states of inertia. Einstein was incorrect when, after proving that no real physical state of inertia was possible, he nevertheless asserted that planetary and astral motion could be accurately described by measurable states of relativity. One real Universe can only ever be eternally appearing in one real state, once. Everything moves at the one Universal constant "speed" of light once. I know that scientists claim to have experimented with laser beams and motion detectors and have proved that a fabricated light can move through a vacuum tube at approximately 186,000 mps, but they never took into consideration the fact that the laboratory they conducted their experiments in was travelling at the Universal constant "speed" of light at the time of the experiments as were the scientists who conducted the experiments. There is no such a thing as mathematics. Reality does not have a plurality of anything, real or imagined.

Dear Members,

I am new to the forum and going through previous essays and topics. My take on "The limits of mathematics". Mathematics is a language and a beautiful one. But we have been carried away by this beauty. Languages can be used to tell the TRUTH as well as LIES and LIES can be as beautiful as truth, that is why we all watch movies and listen to bedtime stories. For example, that unicorns and fire spitting dragons can be well described does not make them exist.

The trouble today is that the physics establishment is unwilling to recognize this limitation of mathematics because of its beauty... "The philosophy seems to be if it is beautiful mathematically, it must be beautiful and exist in physical reality". However, Nature is an unwilling accomplice hence the multitude of paradoxes requiring 'renormalization' (e.g. in quantum field theory) and other mathematical tricks to resolve (see solutions to Zeno's paradox and the intuitive assumptions that have to be forgone, see http://www.iep.utm.edu/zeno-par/ which I referenced in my essay entry below.

As another example, mathematics can be used to divide space and obtain 10-100m. (I am using this figure since Member IAN DURHAM want to avoid the use of infinity). But we are now confronted with evidence that physically this may be an impossibility beyond the Planck length.

Mathematically, you can have a "line without breadth" but Physically this is impossible as such a line will not be real. Therefore, if we want to know 'when and where did mathematics diverge' and if we desire to move nearer PHYSICAL reality, Plato's mathematically correct and possible zero-dimensional points must give way to the physically possible and real MONAD of the Pythagoreans, Leibniz, etc.

See my entry in the 2013 Essay Contest, On The Road Not Taken, http://fqxi.org/community/forum/topic/1764

The limitations of maths when applied to cosmology and physics in general depend on having the correct context in which to apply the mathematical analysis. The use of maths works well when we have a clear description of the physical assumptions and the physical model which determines the formulae to use and the way in which they are applied.

The problem with cosmology is that the evolution of the universe moves forward in time so we aim to work out what happened in the early universe by observing what we can detect today. It is possible to reverse the direction of time in the physical model but we cannot be sure that the evolution did not involve some major transition.

It is better to make certain initial assumptions and then analyse how we would expect the universe to evolve from those initial assumptions. The big bang model works up to a point but it has been necessary to introduce adaptations to make current observations fit with the model. These adaptations include the idea of inflation, dark matter and dark energy. It is not a limitation of mathematics which is the problem it is the choice of initial assumptions.

There are certain key assumptions in the big bang model such as the cosmological principle which has made the mathematical analysis possible but which may not be valid. Also, the question of whether the universe is infinite or finite and if finite does it have a spacetime boundary. The current view is that the universe is finite with a time boundary at time zero but no space boundary. This no boundary hypothesis has been justified by others on the basis that space is a three dimensional surface in a four dimensional space but then having said that they go back to an analysis using a spacetime of three space dimensions and one time dimension. This is inconsistent.

So if we take the basic assumption that we only have three space dimensions and one time dimension then there must be a space boundary or more correctly a spacetime boundary. Now if we start with this assumption we can use existing maths to describe the evolution of the model using primarily the ideas and equations of the general theory of relativity.

The evolution of the universe

The same approach can be applied at the other end of the size scale by challenging the initial assumptions and then applying mathematical modelling in a revised physical world view.

The unification of physics

The nature of mass

Richard

Here are poetries for a humble Review request ,and where as, and without ado : then

from the wild, circus of blatantly-forgotten-advocacy and wistful-proto- abstract representation, . . . Notice for now and/or take in: the naïve visual- philosophic- material as presented by,

http://vixra.org/pdf/1308.0091v7.pdf

and since it is natural , at such out-of- place introductions to procrastinate-, and plainly, just ask why I have addressed for your entertainment ,also a few, personal motivations for such instinctual delay ...

First of all I couldnt possibly answer that myself ; so regardless then .. , would such, arrive from :: ? uninvited urges, to make sense of things, tend in .. Apparent commonality and specious weakness to troll ego, into thoughts, or simply-another-tiny-Insight via the lights of fires long forgotten, or to futures,were they should never arrive;Creating then, satiation and an impulsive cultural cliffing to the untried, or more so, an overall ingrained-frozen-tribal hierarchical rant and warning to safety !

hmm do you already know enough ..? , it would be understandably sad , but maybe ,

You ,never the less, could have simply asked, what is this enclosed [thing],about well I remember when it started, long ago it was about a forbidded, varied illusion of mathematic Time. But for the present, I have grown more humble.and would leave such mystic interpretations to you,as mostly now: its left as an easily demonstrated standard construction and then pre-existent [interior-structure] for the real-numbers;and so ,is , in a sense .. uninterpreted ..hmm

or maybe one could have asked, what do you want from me Impudently and initially, if so, here finally, I would believe, it must be obvious :: if you would be so kind, to take some time, to read it or skim over it , and afterwards perhaps maybe [process this message forward] ; why

. . . . . . . . .

Ive honestly begun to wonder how these structural-thoughts are viewed from an other perspective,as in most of the world, such infinitesimal -or- pointed-set-conversations are curiously stunted or found in the-uncharted references of silence. So maybe you, or again then you might know someone, who could suspect and/or be the right individual to be interested in, or respond with augmenting constructive input or fathom some co-essence, to such instant-like terminal-things and additionally then, I feel that.. this work might then ,find, some unbiased appropriate review,maybe on: arxiv.org (but it seems, I need to woo, an endorsement for that..also-then-painfully-a-hint),

* Lastly, I have looked long and hard for previous work, which by a classic approach, completes the interior of [the Real-number-system] ; since it is such an obvious ,and small step, and could vastly reduce the necessary work as presented or then ,as such, would simply speed and improve the process (by evolutions in at-a-distance like-kind collabortion) . Where hopefully

lucky in return, you may have also gained ,here, some interesting experience, an awareness; and a few beginnings to an- apparent and yet easily passed-over treasure: ie. maths of the moment.

first though the associated work

http://vixra.org/pdf/1308.0091v7.pdf

again really:: I look forward to any-and-all of your input on this outlook, where in the likely and prosaic futures of what seems to me as a yet strange and thoroughly endless complete silence of response, of course, and then sincerely , I hope all remains well for you. -pw

Hi Tim,

It is obviously correct to me except that it doesn't go far enough. But, before I say more, I would like to see if there are any other response. Also, were examples of "a priori knowledge" given in the source for your quotes? Thank you.

James Putnam

Hi James,

I look forward to your further response. Regarding your question, all the quotes were intended as "scare quotes", not direct attribution (sorry for the confusion). To be sure, the argument is certainly not original to me, and I do have a favorite source. And yes, my favorite source does focus on (a) particular example(s) of a priori knowledge, but I thought it best to try to start myself off here as general as possible. If you don't mind, perhaps I will take after you and wait to see your (and any other?) response before muddying the waters further.

thank you,

tim

Tim Rappl,

Hello again. I had my attention drawn elsewhere. Sorry for the wait. I am posting a response below. It isn't about formal logic or mathematics. It represents my perspective on the extent of "a priori knowledge". I will expand it into an essay contest entry and post it perhaps this week. I hope you find it worth your while. Its main thrust is that the evidence for the existence of innate knowledge is provided by the empirical evidence of physics. Not by theoretical physics, but, certainly by empirical evidence:

We talk and write and, because others appear to learn from us, we think that we are teaching them. It is believed that learning comes from the outside world. However, what we receive from others are only signs. The teacher and student are exchanging only photons. Photons are the particle nature of electromagnetism. The ongoing bombardment of a storm of photons does not deliver knowledge. That rush of wildly mixed innumerable bits of data doesn't teach us. Who then is our teacher?

The photons come like a turbulent wind. They are signs of events occurring in the outside world for which we must recognize meanings. The photons are not the events. They are tiny bits of data about changes of velocities of particles of matter that have participated in the events. Individual photons signify tiny pieces of a complex whole. It is their arrangements that signify events in the outside world.

Those ever and always unique fleeting arrangements are what we have for guidance. We, subconsciously, try to fit meanings to them. We subconsciously choose from our innate knowledge the meanings that seem to best fit the photonic evidence for events of the outside world. It can often be easy to make correct associations. When our eyes are 'observing' events, our subconscious mind is working wonderfully well at telling our conscious mind what we are probably seeing.

An incredible amount of data is being analyzed faster than we can see. This ability could not have been acquired by repetition. There has never been repetition. The photonic arrangements are never repeated. The signs are always new. We can pick out patterns in those arrangements only if we already know those patterns. We absorb the photons. We pick out the likely patterns. We do this instinctively.

Instinctive behavior is evidence for the existence of innate knowledge. Instinct means that we have the knowledge necessary to carry out an act without having previously been made aware of it. It originates from inside us. Instinctual knowledge is innate. That innate knowledge is genetically delivered to us.

We do not learn meanings from signs. Words, spoken or written or, what both are physically received as, patterns of photonic bits of data, are only signs. We provide the meanings. Knowledge of meanings must already be available to us before signs can point us toward them. Since we successfully interpret the photon storm, we can know that knowledge of meanings is innate.

James