Friday, August 12, 2011

Quantum Boolean algebra instead of Boolean algebra?

Mathematical logic relies on the notion of Boolean algebra, which has a well-known representation as the algebra of sets which in turn has in algebraic geometry a representation in terms of algebraic varieties. This is not however attractive at space-time level since the dimension of the algebraic variety is different for the intersection resp. union representing AND resp. OR so that only only finite number of ANDs can appear in the Boolean function. TGD inspired interpretation of the fermionic sector of the theory in terms of Boolean algebra inspires more concrete ideas about the the realization of Boolean algebra at both quantum level and classical space-time level and also suggests a geometric realization of the basic logical functions respecting the dimension of the representative objects.

  1. In TGD framework WCW spinors correspond to fermionic Fock states and an attractive interpretation for the basis of fermionic Fock states is as Boolean algebra. In zero energy ontology one consider pairs of positive and negative energy states and zero energy states could be seen as physical correlates for statements A→ B or A↔ B with individual state pairs in the quantum superposition representing various instances of the rule A→ B or A↔ B. The breaking of time reversal invariance means that either the positive or negative energy part of the state (but not both) can correspond to a state with precisely definine number of particles with precisely defining quantum numbers such as four-momentum. At the second end one has scattered state which is a superposition of this kind of many-particle states. This would suggest that A→ B is the correct interpretation.

  2. In quantum group theory the notion of co-algebra is very natural and the binary algebraic operations of co-algebra are in a well-defined sense time reversals of those of algebra. Hence there is a great temptation to generalize Boolean algebra to include its co-algebra so that one might speak about quantum Boolean algebra. The vertices of generalized Feynman diagrams represent two topological binary operations for partonic two surfaces and there is a strong temptation to interpret them as representations for the operations of Boolean algebra and its co-algebra.

    1. The first vertex corresponds to the analog of a stringy trouser diagram in which partonic 2-surface decays to two and the reversal of this representing fusion of partonic 2-surfaces. In TGD framework this diagram does not represent classically particle decay or fusion but the propagation of particle along two paths after the decay or the reversal of this process. The Boolean analog would be logical OR (A∨B) or set theoretical union A∪B resp. its co-operation. The partonic two surfaces would represent the arguments (resp. co-arguments) A and B.

    2. Second one corresponds to the analog of 3-vertex for Feynman diagram: the three 3-D "lines" of generalized Feynman diagram meet at the partonic 2-surface. This vertex (co-vertex) is the analog of Boolean AND (A∧B) or intersection A∩B of two sets resp. its co-operation.

    3. I have already earlier ended up with the proposal that only three-vertices appear as fundamental vertices in quantum TGD (see this). The interpretation of generalized Feynman diagrams as a representation of quantum Boolean algebra would give a deeper meaning for this proposal.

    These vertices could therefore have interpretation as a space-time representation for operations of Boolean algebra and its co-algebra so that the space-time surfaces could serve as classical correlates for the generalized Boolean functions defined by generalized Feynman diagrams and expressible in terms of basic operations of the quantum Boolean algebra. For this representation the dimension of the variety representing the value of Boolean function at classical level is the same as as the dimension of arguments: that is two. Hence this representation is not equivalent with the representation provided by algebraic geometry for which the dimension of the geometric variety representing A∧ B and A∨ B in general differs from that for A and B. If one however restricts the algebra to that assignable to braid strands, statements would correspond to points at partonic level, so that one would have discrete sets and the set theoretic representation of quantum Boolean algebra could make sense. Discrete sets are indeed the only possibility since otherwise the dimension of intersection and union are different if algebraic varieties are in question.

  3. The breaking of time reversal invariance is accompanied by a generation of entropy and loss of information. The interpretation at the level of quantum Boolean algebra would be following. The Boolean function and and OR assign to two statements a single statement: this means a gain of information and at the level of physics this is indeed the case since entropy is reduced in the process reducing the number of particles. The occurrence of co-operations of AND and OR corresponds to particle decays and uncertainty about the path along which particle travels (dispersion of wave packet) and therefore loss of information.

    1. The "most logical" interpretation for the situation is in conflict with the identification of the arrow of logic implication with the arrow of time: the direction of Boolean implication arrow and the arrow of geometric time would be opposite so that final state could be said to imply the initial state. The arrow of time would weaken logical equivalence to implication arrow.

    2. If one naively identifies the arrows of logical implication and geometric time so that initial state can be said to imply the final state, second law implies that logic becomes fuzzy. Second law would weak logical equivalence to statistical implication arrow.

    3. The natural question is whether just the presence of both algebra and co-algebra operations causing a loss of information in generalized Feynman diagrams could lead to what might be called fuzzy Boolean functions expressing the presence of entropic element appears at the level of Boolean cognition.

  4. This picture requires a duality between Boolean algebra and its co-algebra and this duality would naturally correspond to time reversal. Skeptic can argue that there is no guarantee about the existence of the extended algebra analogous to Drinfeld double that would unify Boolean algebra and its dual. Only the physical intuition suggests its existence.

These observations suggest that generalized Feynman diagrams and their space-time counterparts could have a precise interpretation in quantum Boolean algebra and that one should perhaps consider the extension of the mathematical logic to quantum logic. Alternatively, one could argue that quantum Boolean algebra is more like a model for what mathematical cognition could be in the real world.

For details and background the reader can consult either to the chapter Physics as Generalized Number Theory: Infinite Primes of "Physics as Generalized Number Theory" or to the article How infinite primes relate to other views about mathematical infinity?.


At 10:42 AM, Blogger Ulla said...

What do you think of these? Horava-Lifshitz gravity, a minimum scale and N^2 (for short distances), + a modification of Newton mass.

I learned that c is an ad hoc assumption made by Einstein.

At 1:30 PM, Blogger ThePeSla said...


c or h for that matter as an idea of measure (that is concerning finite measure is possible as inertia) and we give general relativity that sort of absolute time reference) is not fundamental. But we can so modify the values involved to fit more up to date physics. All is in this sense an ad hoc assumption. But is it not amazing the models work over a wide universe regardless of who adds what or not to it all?

Einstein and Newton share this same primitive idea of mass. Newtons absolute time was then distinguished from Newton's relative time.

The PeSla

At 6:25 PM, Anonymous said...


c is definitely not an ad hoc assumption made by Einstein! What Einstein introduces is pseudo Riemannian geometry and possibility of light-like curves and geodesics along which distance vanishes. c is just a numerical constant in particular coordinates guaranteeing that dtau=cdt and spatial coordinates dx^i have same dimension In some coordinates it would depend on position but this is not essential. The geometric meaning of Lorentz boosts (motion with finite velocity v<c is as a hyperbolic rotation. Something very non-trivial. There are many bad mis-understandings related to the meaning of c: the variable c model of Riofrio is a good example in this respect.

Special relativity differs from Newtonian relativity because in the latter time is absolute and same for system and system moving with respect to it.

Horawa's gravity assumes 3-D insted of 4-D general covariance and means assumption of special time direction in conflct with special relatvity. One starts from Einstein's equations in so called ADM decomposition and in these special coordinates introduce an ad hoc modification.

This tinkering is not of course in spirit in general relativity and relies on no principle. As a crime this would be comparable to a "slight improvement" of Chopin's Fantasie Impromptu: I do not believe that anyone has done anything like this;-). On basis of what Lubos has reported Horawa's idea was immediately shown to fail. Horawa modification represents to me one of the many short lasting media fashions which make me so dissappointed on the publication policy of New Scientist;-).

The possibility to avoid string theory might be represented as a justification for Horawa's theory. This kind of ad hoc theory is certainly not a solution to anything. How to modify general relativity without giving up general coordinate invariance and equivalence principle and solving the problems related to the definition of energy and momentum? This is to me the real question.

Somehow one must combine special relativity predicting classical space-time symmetries and general relativity providing excellently working model of gravitation. It is needless to continue further;-).

At 12:05 AM, Blogger Ulla said...

It took many years before Einstein could justify his assumption, which from the beginning was ad hoc.

Horava has the spacetime sliced into membranes too, like the Big Book. (depending on 2-D?)

This question of inertia-change is the same as determinism - non-locality? Of course.

I am definitely a bit stupid after all :) Why is one modification good and the other disgustful?

At 3:05 AM, Anonymous said...

Einstein's general theory of relativity is a theory based on few general principles from which everything follows. General Coordinate invariance and Equivalence Principle and geometrization of gravitation.

Horawa takes Einstein's equations in very special formulation in very special coordinates and makes completely ad hoc change in them.

Determinism and locality are notions which do not have much to do with the notion of inertial/ gravitational mass.

The slicing to membranes emerges just because 4-D general coordinate invariance is given up. One can of course obtain slicing dynamically but this is totally different thing.

At 11:12 AM, Blogger Ulla said...
my source. Jimena Canales.

From 1905 to 1907 Einstein’s work changed from being an investigation of time signals to signals in general, but by 1910 Einstein reframed his research in yet another way. Its implications for signaling were described as a “consequence” of a much broader physical theory—and a profoundly counterintuitive one at that. Conditions necessary for sending or receiving signals were what “follows immediately” from his theory—not its starting point.

Contemporary technological limitations in signaling speed were the reason why the speed of light appeared as a constant in Einstein’s theory. That is why “we are driven to the convention that light, as our quickest system of signals, is moving with uniform velocity.” [is not that an ad hoc statement?]
While Whitehead understood and accepted the radical conclusions of Einstein’s theory with all its paradoxes, he nonetheless turned the physicist’s argument completely on its head. Einstein started his argument by insisting that the constancy of the speed of light was a universal fact of nature; Whitehead started his by noting that it was, to that day, the quickest way of sending signals.

Many considered that the value of the speed of light, c, often expressed in kilometers per second
was so technical and so lacking in elegance (it did not even come close to being a nice round number or integer) that it was unfit to parade as a universal absolute. Einstein disagreed. Even up to the last years of his life, Einstein combated critics by stressing how the seeming arbitrariness of the number c could be eliminated. If the unit of seconds from the relativity equations was replaced by “the time in which light travels 1 cm” it could be made to equal one. the constancy of the speed of light appeared much more natural, logical, necessary and universal.

I will not occupy the blog with these old debates. Horavas first draft.
This theory, which at short distances describes
interacting nonrelativistic gravitons, is power-counting renormalizable in 3 + 1 dimensions. Symmetry (and GR) is emergent! It is not proved, but still one can have an opinion. With the later modifications it looks better? I shall look at Lubos text.

At 11:42 AM, Blogger ThePeSla said...

Matti, Ulla,

Glad you saw my comment in the greater context of our search for a more general theory.

Now, consider this, no signal can be sent faster than light (save perhaps once a path is established as in Penrose's idea of quantanglement And what is that signal but some form of information bits?). So as Shannon observes a signal is information that contains meaning, thus we are describing a certain entropy of a message.

Now I too have thought about Rio Frio's desire to see early (quasars) as evidence of the slowing down of light. I have no problem with this conclusion really but it may exist as a principle in other forms or other principles may explain things.

For me a quasar is a creative object or the result of one, but one could say (since she points out it was so close to some beginning of things another principle could be involved) that the first value of c was as if it was nearly infinite to so slow down. (to what? nearly or absolutely zero velocity?)

Determinism and locality, absolute chance as non-linear and not clear in direction (thus also a true unitary theory as far as quantum like grounding of continuity needed goes) and locality or non-locality, in general what grounds our ideas of coherence, and if we desire as if a fourth concept Inertia (be it mass or gravity) somewhere has much to do with mass- not what the standard theory or GR seems to think as it is not general enough.

What does c slow down relative to?

Clearly, the invariant hyperbolic rotation or boost is a z axis or what some may see as a limitation of such symmetries of space as we think it merely 3 or 3+1 dimensional. Kea has some objects that seem to suggest this sort of thing in that terminology.

But I am still not quite sure what you mean, Matti, by such a distinction of the Boolean notions yet I did imagine within the same framework of space that as sub notions there are physical things and processes we can so distinguish. (see my post today on toward a philosophy of physicality) the issue of which is precisely your statement of the way or problem of solutions.

I am also not clear on what it means to slice membranes or if these are on the same level as TGD.

And since when does c have an exact value but one we assign it to some degree of accuracy as the standard?

Who the heck is Horawa?

Anyway, today's fashion may turn out permanent- ultimately Lubos shows a deep indeterminacy at the heart of things which does after all make simple string theory a possible concrete foundation (even when Higgs fields as such are not) so why expect they to give mass values in any one of the wide landscape geometries? Can a theory be so abstract it does not apply?

But yes, modification of a theory can mean it is out of touch.

The PeSla

At 4:36 PM, Anonymous Anonymous said...

Sorry, silly elementary noob question:
How many pi is c?

And another thing, the reduced Compton wavelength (lambda), which according to wikipedia states:

Lambda/2p = hbar/mc

What perplexes me, isn't wawelength divided by circumference of a circle by definition 1? In which case c = hbar(h/2pi)/m

Now again, what is the number theoretical value or essence of hbar?

PS: what does "CP_2" stand for and mean? Sorry again for silly noob questions... :)

At 1:01 AM, Blogger Matti Pitkanen said...

Riofrio's varying c could make sense in the context of general relativity only in the sense of Robertson Walker cosmology for which c^2dt^2 in line element is replaced so that c becomes t-dependent and approach zero at the Big Bang, just the opposite for Riofrio's proposal. This would not be a Big Bang anymore but something totally different.

Anonymous asked some questions: I answer to those that I understood. hbar is quantization unit of angular momentum -quantization is what is highly non-trivial and non-classical- and it appears everywhere in quantum theory. As such hbar has no number theoretic meaning.

CP_2 is so called complex projective space of complex dimension 2. The notion of complex space and projective space would be needed to understand what is involved. Wikipedia gives details, which are however difficult to explain without needed mathematical context.

At 5:15 PM, Anonymous Anonymous said...

The Value of pi and The Speed of Light:

At 6:53 AM, Anonymous Anonymous said...

Further notes on c expressed in terms of pi:

First, math being Greek to me, my mathematical understanding and power of expression is limited to most simple forms of lines and circles.

Lets start from couple axioms derived from Matti's theory,
1) that I am a conscious entity the size of earth,
2) that my Moment of Consciousness (MC) aka "Self" in standard metric is 0,1 seconds and
3) that ligth circumferes Earth (or I) 7,5 times per second, ie, 0,75 times per MC.
4) certain axioms of Number theory to produce the 1-dimensional Real line
5) axiom of "quantum jump of Imagine" from Real line to square root of -1, defined as it was originally and etymologically justified, namely quarter of circle's circumference from the Real Line into new dimension (D+1), counterclockwise from 1 on Real line


to complex number (0, i), implicating a 2-dimensional Complex Plane or Sphere.

Now follows the conjecture which I name the "Grounded Conjecture of c, pi and Moment of Consciousness", derived from axioms previously stated:
1) Imaginary number i equals (pi*Real line radius (R))/2
2) hence, speed of light c is three quarters (0,75*circumference of I, conscious being the size of Earth) of the circumference of the circle implied by jump to i, in other words 3i - or - 3 * (pi*R) / 2 - or - 1,5piR. Let's remember that 3i equals Moment of consciousness that closes the initial jump i from Real line point 1 into full circle back to 1.

Result 1,5piR is highly non-trivial, as 1,5 leads to point ½ on the real line beginning from point -1, namely the Real part of the critical line and non-trivial zero's of Riemann's hypothesis.

To put it short, this Grounded Conjecture states that c = Moment of Consciousness 3i = 3*(piR/2).

Of course, 3*(piR/2) when R=1 is multiplied by Earth's Radius in metric system gives about 30 000, which multiplied by original definition of metre (which was 1/40 000 of Earth circumference) gives about 300 000 000.

Actual measurements in terms of Real numbers are of course more wobbly and complex, reminding us of what I call the Axiom of Love or Axiom of Creation, which states that more (complexity) is better, And which more or less equals Riemans Hypothesis... :) <3

At 8:55 AM, Blogger Matti Pitkanen said...

Dear anonymous,

c is *dimensional* constant in standard units. With a suitable choice of units onw has c=1. It does not make sense to assign to c any "deeper meanings" since meter and second (or any other choices) are just accidental choices of length and time units.

In the case of *dimensionless* constants such as fine structure constant or ratio of proton mass to Planck mass situation is different and it makes sense to ask why fine structure constant is about 1/137 and proton to Planck mass ratio is about 10^(-38).

At 10:59 AM, Anonymous Anonymous said...

Dear Matti,

could you please try get of your High Horse for a moment and level for a while with silly old I? :)

If you read carefully you will see that the conjecture about relation between c, pi, i and moment of consciousness is not dependent from "meter and second or any other choises" or sizes. It's purely number theoretical, based on the "deeper meaning" that the most basic number theoretical constant pi is present in all curved shapes - lines and spaces, starting from the circle.

Sure enough, the conjecture was inspired on the empirical coincidence that for the duration of what you say is Moment of Consciousness for observers like us (size scales like us), ie. 0,1 seconds in terms of the accidental choises that we are accustomed to, a foton travels 3/4 of the circumference of Earth. Which you claim is the size of the magnetic body of beings like us.

How do you yourself derive or base 0,1s as moment of consciousness and size or Earth as the human scale magnetic body? And what do you yourself make of the coincidence that during a moment of consciousness light travels 3/4 of the circumference of the consciouss entity in question? And second question, how does c, dimensionless or dimensionfull or what ever in any kind of infinity, relate to scalable hbar and dark matter hierarchy?

I am a Strange Loop, as Hofstadters booktitle states, and even though hyperoctonions, infinite primes and all the other complexities of the jargon of a theoretical phycisist and mathematician are all fine and dandy, this Strange Loop finds it's conjecture about itself very simple and hence beautifull, Not "even though" it can be expressed in simple terms of lines and circles, but because such simple geometrical dynamics are not incomprehensible even to laymen without formal education in theoretical physics.

At 2:21 PM, Blogger ThePeSla said...


It takes a very high horse to make high grade horse hockey- so why the remarks as if you are in the know so feel morally justified behind a mask (save perhaps your spelling of foton)?

Hofstadter is not kind to his students and his strange loop idea is wrong or barely relevant even from a physics viewpoint.

This is a post that asks us to consider some deeper points of logic (of which Hofstadters is a little dated if it applies anymore at all).

Does pi not arrive independently from statistics and not just a circle diameter ratio? Can given any 9 digits of it we predict the next 9 in hexadecimal? Sorry the concept of numbers (Pitkanen's strength and originality in my opinion) as hierarchies or such concepts as animal magnetism are popular when we do not fully understand magnetism.

You seem to have no idea that in the physical world as finite we can have a duality of circles and lines drawn beyond loops of our comfort zone of infinity. So your ideas on this are not deep enough for even a layman to understand.

A true intellectual, honest scientist, and layman should try to imagine there are things he may not know (outside of Godel, Escher and Bach) of which he should do the hard work of showing if it can be done why he makes such comments.

The second question I give you this: the key is how things connect in the wormhole like idea between organic molecules like DNA of which from a metaphysics standpoint one could certainly say consciousness and its asymmetry of necessity exceeds the velocity of light. Certainty you have experienced the hand faster than the eye?


At 2:46 PM, Blogger Santeri Satama said...

The conjecture I tried to sketch above is of course closely related to Euler's identity:

where nepers number ^ (i * pi) = -1 on the Real line, half a circle from 1 to -1.

The length 1,5 that the - perhaps idiotic - conjecture about relationship of c and pi strangely produced, when added to -1 of Eulers identity, leads to the real part of the critical line of Riemann hypothesis.Does that constitute a proof of the Riemann hypothesis? Perhaps, sort of, and if so I will gladly share the reward sum with Matti: half for me and half for you... ;)

At 5:26 PM, Blogger Santeri Satama said...

Dear Pesla,

I'll try to answer your questions concerning duality etc. with another conjecture, related to the axiom 'more complexity is better'.

From two feeted vertical position to getting on a High Horse, basic human tendency seems to be upwards, to the top of the hill. This is seen also in the choise of drawing quarter circle of imaginary unit and half circle of Eulers identity over the real line, not under. Positive is up, negative down, in most basic spatiotemporal evaluations.

And this is of course perfectly natural for infantile mammals like us, our first and most basic instinct urging us to climb to the top of the tit in search of the nipple from where the sweet flow of white pure energy comes and keeps us alive.

King of the hill, top of a pyramid hierarchy, we all have experience of these games.For mathematicians the memory image of the nipple (n*i*pi²*lambda*e?) is most perfectly represented by the imaginary unit i, offering math-shamanistic axis mundi from ground level to bird eye view from infinity that enable complex projections from spheres to planes and vice versa etc etc.

So, with our obsession with tits (with nipples on top) and our fear of bumbs, IMHO the most real beauty of Eulers identity, whether one draws the semicircle over or under the real line or both, is the +1 radius to the center of the circle, to the heart of our Mother Earth.

At 8:17 PM, Anonymous said...


if there is some deep connection between c and pi, it appears only when very special units of velocity are used. m/s as a unit for velocity should represent this kind of very special velocity. But this cannot be the case bow since relativity prevents this kind of velocities. There should be a special inertial system in which m/s would be special kind of velocity. This would be resemble ether hypothesis.

In condensed matter of course sound velocity is something special but it is velocity in the special inertial system defined by condensed matter taking the mathematical role of ether.

The simple rule of thumb is that only the values of dimensionless constants can have deeper meaning.

At 8:54 PM, Anonymous said...


thank you for questions. I try to answer.

1. Concerning the role of Earth's magnetic field and relevance of Earth's size scale for consciousness the basic observations are following.

a) EEG frequencies correspond to wavelengths of order Earth radius. Schumann frequency 8 Hz corresponds to Earth's circumference as wave length. Frequencies in this range are known to have effects on vertebrate brain. This was the original empirical input leading to the idea that the function of EEG is to make communications between magnetic body and biological body possible.

b) The size scales for magnetic flux quanta of magnetic body of Earth follows directly from what we know about magnetic field of Earth. In first approximation it is dipole field whose dipole strength we can estimate. The strength of B at Earth surface varies somewhat from poles to equator, this makes as variation in cyclotron frequencies.

c) By the way the value of "endogenous" magnetic field assignable to the flux tubes of magnetic body associated with vertebrate brain .2 Gauss: 2/5 times smaller than the nominal value .5 Gauss of Earth's B. Schumann resonance frequencies depend in the first approximation only on Earth's radius and c.

2. You ask also about the relationship of c to hbar. By a proper choice of units believing that there is no ether like preferred inertial frame chooses units so that one has c=1. This makes formulas much simpler and easier to remember;-). One can choose the units also so that hbar=1 holds true. If one takes seriously the message of TGD then only hbar_0=1 can be assumed and hbar=n, n=1,2,… holds true in this convenient unit system. c and hbar appear
in fine structure constant and its expression becomes very simple: alfa= e^2/4*pi when hbar_0 =1 and c=1 drop out.

To be continued....

At 8:56 PM, Anonymous said...


continuing to answer to your questions....

3. What is the geometric span for a moment of consciousness depends on the level of the hierarchy of conscious entities. The reason is that moments of conciousness (identified as selves in the most economical view) form a fractal hierarchy and so do their spans.

a) Human magnetic body has fractal onionlike structure. Each layer of the onion corresponds to some characteristic
length scale and corresponding wave length is assumed to correspond to a what might be called scaled variant of "EEG" used by a particular body part to send sensory data to its magnetic body and used by that magnetic body to control the body part.

Body part can mean DNA, cell membrane, cell, organelle, organ, organism, population, even species if one accepts Sheldrake's ideas. The higher the level in this hierarchy the larger the corresponding magneic body. There is entire hierarchy of frequency scales beginning from at least at visible frequencies about 10^(14) Hz and ending at least up to frequency scale defined by life-time. For the frequencies below 10^(12) Hz (roughly) the energy of photon is below thermal energy at room temperature for ordinary hbar but if hbar is larger than its minimal value it is possible to get larger energy E=hf.

This is very essential for having quantal effects for low frequencies: they are very essential for memory realized as communications with geometric past: energy of very low frequency photon must be above thermal energy at physiological temperature.

b) A natural proposal is that causal diamonds (CDs) serve as geometric correlates of moments of consciousness/selves at imbedding space level: if so a natural span for the contents of consciousness corresponds to temporal distance between the tips of CD in the rest frame of CD. One can assign this kind of CD to any elementary particle. The proposal motivated by p-adic mass calculationsis that this temporal distance is a multiple of the time scale tau=R/c defined by the radius of CP_2- certainly a fundamental length scale.

For electron this temporal distance is .1 seconds which happens to correspond to what is identified as duration of moment of sensory experience. There are also other scales and it might well be that many elementary particles correspond directly to time scales of conscious experience (I have discussed this for quarks for which time scales would be shorter). Hierarchy of Planck constants gives rise to integer multiples of these time scales.

c) Many other time scales are relevant to consciousness: cyclotron frequencies, Schumann frequencies, etc..

At 11:39 PM, Blogger Ulla said...

Good that my silly comment got that big response. Sometimes we need go back to the simple questions. Quantum Boolean algebra has very much to do with this :)
The arrow of time. Fill in all the quantum effects, and susy.

One more silly question. c=hbar=1
What is the relation between c and hbar?

Time scales = size scales. Time is an operator. Why? Because of c.

Lorenz invariance must be derived. And we know it is not always true, because of c or time :)

At 11:50 PM, Blogger Ulla said...

By a proper choice of units believing that there is no ether like preferred inertial frame chooses units so that one has c=1.

What would the quantum 'inertial' soup be called? And how would it invoke? It determine the scalings and indirectly also time through size?

I must confess I have difficulties understanding your explanations, which certainly are good.

At 12:00 AM, Blogger Ulla said...

"There should be a special inertial system in which m/s would be special kind of velocity. This would be resemble ether hypothesis."

SPEED of light is such? SPEED.

At 12:10 AM, Blogger Ulla said...

"c and hbar appear in fine structure constant and its expression becomes very simple: alfa= e^2/4*pi when hbar_0 =1 and c=1 drop out."

Santeri asked about the relation of c and pi. It has a condition. number 1. What happen if the base are other numbers?

And how does this relate to the flatness problem of Universe?

At 4:25 AM, Blogger Santeri Satama said...

Thankyou, Matti and Ulla.

I feel that the search for "deeper meaning" or some Great Secret is leading astray, since the number theoretic motivation - reduction of physics to number theory - leads to seaking a purely number theoretic metric that is not dependent from meters and seconds and such. My conjecture can and should be taken as a proposal to adopt pi (on the condition of number one and Eulers identity) the number theoretical basic unit of length etc. For example, wavelength sin(x) = 2pi (

"EEG frequencies correspond to wavelengths of order Earth radius. Schumann frequency 8 Hz corresponds to Earth's circumference as wave length."

Can you express those correspondences without the messy unit Hz depending from the messy unit second, using pi as basic measure - for example giving the Earth radius value of 1?

At 5:02 AM, Anonymous said...

To Santeri.

I could of course introduce the the temporal distance between tips of electron's CD as a fundamental time unit and define unit of frequency as inverse of this: 10 Hz, the alpha frequency and fundamental biorhythm.

Number theory and geometry do not exclude each other by any means. The vision is that the Kahler geometry of infinite-D world of classical worlds (WCW) is unique from its mere mathematical existence. That infinite-D geometries typically fail was discovered by Freed for loop spaces, which are must simpler objects.

WCW must decompose to a union of symmetric spaces for which all points are geometrically equivalent: this is extremely powerful condition and implies maximal symmetries.

These infinite-dimensional symmetries must have very special meaning and the natural guess is that they have interpretation in terms of number theoy. Classical number fields are indeed a key ingredient of the theory and allow interpretation of standard model symmetries number theoretically. Physics as infinite-D geometry and Physics as generalized number theory visions become more or less equivalent.

At 6:33 AM, Blogger Santeri Satama said...

Timeo danaos et dona ferentes - beware the Greeks bringing gifts. Was it roughly c, 300 000 000 monetary units, that Greece promised as guarantee for a loan from Finland ;)?

Greeks gave world the pi, abhorrence to Pythagoreans. It took a while to prove that circle cannot be squared, because pi is not an algebraic number but transcendental.

Shouldn't an algebraic number theory and geometry be able to jump over the pi-fence to greener pastures or can it and all other irrational transcendentals be just ignored? Even though almost all real and complex numbers are transcental? Should non-algebraic transcendentals (that don't code with p-adics?!) be ignored anymore than the non-coding trash DNA, and if so, why?

Do transcendentals deserve to be called geometrically equivalent points in your vision of symmetric spaces for WCW?

Electrons are friendly, they wave at you. But from which side of the pi-fence, if there is one?

At 8:23 AM, Anonymous said...

Transcendentals in real context are part of the story. p-Adic transcendentals are p-adic numbers with non-periodic expansion in powers of p and infinite as real integers. Also these are unavoidable.

The generalization of number concept means gluing of real and p-adic number fields and all their algebraic extensions together along rationals and possible other common points. This is book like structure.

The intersection of real and p-adic worlds - back of the big book- consists of rationals and also some algebraic numbers even real transcendentals such as e^p. This intersection would correspond to the intersection of matter and cognition and to life. Cognitive representations cannot never give faithtul representation of transcendentals.

Pi is very interesting number p-adically. The numbers exp(2pi/n) are algebraic so that exponents, sines and cosines can be represented in p-adic world in terms of algebraic extensions of p-adics. Pi would require infinite-D extension of p-adic numbers and I do not like it.

One could however start from a geometric definition: 2pi is measure for the length of unit circle. What unit circle could be in p-adic context and what could be its length? Could one define the length in finite measurement resolution as sum of identical lengths of sides of regular polygon with N sides. This would give an algebraic number. One can imagine also other definitions but this looks rather
concrete and understandable and gives something approaching 2pi at the limit when polygon has infinitely many sides.

At 10:40 AM, Blogger ThePeSla said...

Dear Santeri,

Nice name by the way.

I think you may have got the wrong end as to which way the axes goes, the horizontal or vertical.

Sex is a good way to make understandable what to the masses is otherwise obscure physics.

But quantum physicists are drawn to the mountain tops and so fall off sometimes.

But what is a pyramid but crude compression compared with our cathedrals of the flying buttresses and fan vaults?

Where does the worm go in half a higher space apple?

And if we strive for the mother's milk of the mountaintop- is she but the nursemaid or is the nursemaid the mother?

ThePeSla, but a Layman by the way.

At 11:46 AM, Blogger ThePeSla said...

Matti, everyone,

I have some problem with our interpretation of physics as if a too restricted "monological" view.

I mean even where we duplicate (that is our core idea of group duality) some ensemble say at E8, I see not reason to regard the copy as that indicating gravity. There is much more freedom of the relations as such.

So quantum gravity, especially where it strives to find a monlogical relation to the Planck energy as in string theory is not a general enough theory. The idea of these dimensioned or dimensionless choices of our measure makes more sense if we allow concepts like a hierachy of Planck values as TGD asserts.

There is no reason in general to prefer pi or its multiples other than convenience of the formulas of our systems, to force a mathematical principle on the cosmos over other ways to do it.

In a sense there are infinitely more transcendental numbers but we have proven but a few of them. So from a wider view, in these paradoxical matters of singularities all such seemingly non-linear relations of numbers can be set down as algebraic if the information is real and useful.

I think the burning questions of what happens say if we have infinite levels of p-adic numbers will vanish as an issue if we strive to behold a wider ground. But in the meantime we certainly need to develop our methods and systems as best we can.

And the issue of zero point vacua should be considered from a little more relaxation view- not to say that in the world of varying things gravity and mass may on the whole be constant.

It has been said that the ratio of a diameter to a circumference could encode somewhere the encyclopedia Britannica or works Shakespeare could have written. Or anything of some possible cosmic reality. Does this come back after some cosmic year? But how much richer would these sequences interact if in the possibility of differentiation of flat things that as the computing of the next higher dimensional volumes we have to square pi itself?

The PeSla

At 7:02 PM, Blogger Santeri Satama said...

Dear All,

Zero energy ontology/dialectic, holonomic holography, Matrix of WCW reducing to math as we like it and then some WOW!

I cannot but comment on Matti's words "should we take the message of TGD seriously", not only with the perhaps allready worn out and blunt guillotine "no should from is" but even more seriously, why "seriously" and not emmm... in search of a better word, POETICally?

POETIC, as in PO Phosphor-Oxygen Energy*Times i and c - and what not to pi or not to pi (the joke is somewhat lost both in Finnish, English and Finglish, which should bi fine(d))? Etic and ethic criteria for our (POur) map-landscapses asking simply, how would (and do) you feel living in the TOE that you occupy your inner dialogue with, in a rather creative way? Or would you prefare heel? Does your W of WCW-Matrix make you happy and content? A Doctor Feel-Good-God?

Somehow it feels that the irrational transcendentals don't really mind mind, they just lurk and play seek and find and hide. For the mind behind, to end with a rhymend.

At 9:48 AM, Blogger ThePeSla said...


I think we are well beyond the post modern philosophy of science here.

You are welcome to explore with us.

If you want poetry as such see my otherwise

See what happens when you say looped in the holodeck, "Computer, End Program."


At 11:14 AM, Blogger Ulla said...

somehow I think you know Matti well :) Nobody else would dare to talk like that :)

TGD is a symphony, an artwork within the frames of science. There is very much mind. The 'dance' doesn't yet go painlessly. Cognition is very badly researched, even worse than consciousness. Memory we don't know what it is, etc. In fact we have a very primitive picture of everything so far.

For many badly known pictures biology can show the path forward.

I wouldn't mind living in the TGD Universe. So spiritual! And beautiful! And stubborn :)

For arXive - l-adic and m-adic preprints are allowed. Why not p-adics? Still a famous, old profa said - nobody does work with p-adics, and he couldn't understand why not. I told him someone do :) Instead, as Franca and Lubos, they ridicule this branch (and me, but it is maybe another thing. My comments are a bit extra, mildly said :) Poor Matti:))

At 7:22 PM, Anonymous said...

I dare to to speculate that when Higgs leaves, p-adic physics comes. When the time is ripe for a new idea, nothing can prevent its breakthrough as someone has said.

As a matter fact, it is incredible that after 16 years from the first variant of p-adic mass calculations predicting correctly the masses of known particles and also predicting a lot of new physics and even published in arXiv (still possible at that time!) colleagues are still wasting their time by studying extremely complex variants of Higgs mechanism with no hope about what might be called understanding. This is dictatorship of mediocricy.

Next monday (Lepton-Photon 2011) might be the day when they announce about evidence for Higgs around 144 GeV: Phil Gibbs told that he has received a rumor about indications for Higgs with this mass.

It will probably take year or two to learn that it was not Higgs or not. What puts of course bells ringing that this mass corresponds to CDF 145 GeV bump which did not allow identification as Higgs.

Pseudo scalar or scalar?: this will be the question. The decay to two photons is a clear signature for pseudoscalar and could finally demonstrate that pion of M_89 hadron physics is in question (at least to me!) because the decay rate is completely fixed from anomaly considerations. No doubt evidence for other mesons such as rho and omega will be collected. For them evidence exists already and their masses are predicted correctly by TGD assuming 145 GeV mass for M_89 pion.

After this colleagues must accept the possibility that Nature has chosen TGD. No doubt there will be a very painful transition period: not only for Lubos and Franca;-). Probably I will be labeled as a madman also publicly. Those good old fellows who have silenced me for decades will never forgive that I was right.

This is of course only one possible flow of events. Only future will show what really happens.

At 3:54 AM, Blogger Ulla said...

You are maybe mad in the sense you have persisted and continued, in spite of all. I really appreciate it, though you have sometimes thought otherwise. In that sense the stubborness is good ;-)

Your other sensory handicaps (;-)) are, however, not for good, but not even with worst scenario they can be called madness. Humans are not perfect, none.

Envy and humiliation can create rage, a desire to hurt. You only need to think at a marriage ending.

Forgiveness is to release, let the burdens/what has happened go. Open up the hands so the bird can fly away, if it want to, or if you want to. Freedom. But that means not free from emotions, you just let them pass, experience them, you just dance the dance of today :) not yesterdays dance. Emotions are the energy (bad or good) making you dance, as you so cleverly noticed. They are not you, but essential for the movement. They make your guitar sing :) Also the guitar is needed :) No crackpot will manage!

Let us drop yesterdays emotions and recieve todays', as they are, without labels, attitudes, envy, jealousy, fear and other feelings of yesterday. Are scientists more drown to yesterday? The money again?

But I always forget I am talking to deaf ears :)

At 3:44 AM, Anonymous said...

I am of course not mad in the sense that the power elite of science would like me to be. I am not a paranoid, I am not a schizophrenic, I do not have any delusions, I do not hear voices, nor do I experience anything else which would sound like a clinical signature of madness.

The problem is that I happen to be quite an intelligent scientist which ability to have bird's eye of view and see what is there. My technically talented colleagues tend too often to see only their models and their methods.

This far sightedness irritates specialists whose self confidence relies on ability to make tricks that those others cannot. They are skillful jugglers.

Why physics is for me is a continual process of becoming conscious about beautiful ideas is not because I would be somehow special but about giving up the attempt to be Academic Somebody, to be the skillful juggler able to make amazing ricks. The most boring music is composed by virtuosos (Chopin and Beethoven are rare exceptions but they had great soul;-).

At 11:48 AM, Blogger ThePeSla said...


back to the logic questions.

Lately, in seeing more of the extent and depth of your reaching I suspect that you would benefit from my so called quasics view as well I would benefit from the more Lie group ideas of the topologies.

We can of course default it all to binary- for that after all is the structure of such Boolean logics and yes beyond a quantum aspect of it or parallel the (finite or Klein) geometry can revert in the end to more classical logics and not these intermediate steppingstones.

You see, at least we have the sense of beauty in these rarefied realms of physics while it seems that so many play without the reach of Chopin play chopsticks on a grand piano - or sing along to the great karaoke in the sky until the drink kicks in following the bouncing ball faster than the jittery film in their own heads as if he scroll of he player piano were their own.

Are we the designers of such logic systems or within the sense of its design? What of uniqueness as determined by primes as in Godel which is after all a metalanguage rather than say a statement of physicality? But can the same subjectivity of notions not be said of Boole?

Now on newscientist today an article that crudely suggests that most of mass, and the new data may have to redefine mass radically anyway, is in the black holes (the cosmological editor there). But this may be the case in a sense if he means structured singularities.

One does not have to be perfect to recognize perfection in a world where the old logic does not apply and (Ulla) you would like to live in a TGD universe you say- but well, you may be surprised somehow that you already do.

The signs of things simplify equations, the bad or good emotions are merely that and have no ultimate reality fixed beyond the local time of solving things. And the game is that the voiceless and deafness do speak to us as we are driven to hear, discern the message.


At 7:56 PM, Blogger Santeri Satama said...

How advanced is the math that is known - or known unknown?

I just found out e.g. that the Euler-Mascheroni Constant, the limiting difference between the harmonic series and the natural logarithm, is not well known at all - it's not known if it is algebraic or transcendental, or even if it is irrational or rational:

Gamma-constant manifests among other things in dimensional regularization of Feynman diagrams in QFT. Would TGD as generalized number theory be able to predict if gamma-constant is rational or irrational, algebraic or transcendental?

At 8:56 PM, Anonymous said...

To Pesla: the least radical view is that quantum logics with co-AND and co-OR and other co-variants of basic binary operations is relevant for understanding what happens conscious logical reasoning involving problem solving in the spirit that Poirot does it. Mathematician might be happy with ordinary logic.

In any case, it would be nice to have a summary about what it is to to solve a problem? Some kind of classification about different kind of manners to be a problem. I have tried to solve problems whole my life but find myself completely incapable to write this summary. Perhaps this explains the outcome;-).

At 9:46 PM, Anonymous said...

To Santeri:

Thank you for a excellent question. Your problem boils down to the fact that there is no numerical method for telling whether some number is rational. algebraic number, or transcendental. This limitation of numerics would be in turn a restriction of cognition if p-adic view about it is correct.

Transcendentality can be however proven for some transcendentals such as pi? Why this is possible? What distinguishes "knowably transcendentals" like pi from the rest which are able to hide their special nature;-)?

a) Certainly for "knowably transcendentals" there must be some process revealing their transcendental character. How pi and e are proven to be transcendental? The process could be a demonstration that these numbers cannot solutions of any polynomial equation. When I look to Wikipdia about "squaring of circle" I find link to Weierstrass theorem proving that pi is transcendental by this criterion.

b) The basis theorem is that exponents of algebraic numbers are transcendentals (see this). This is something extremely deep and I do not know the proof. The theorem produces huge amounts of "knowingly transcendentals" since any algebraic defines a transcdendental by exponentiation!

c) This theorem allows to prove that pi is transcendental. Suppose on the contrary that pi is algebraic number. Then also i*pi would be such and this would mean that exp(i*pi)=-1 would be a transcendental by previous theorem and this is of course a contradiction. How do we know that e^(i*pi)=-1 holds true. Euler demonstrated this. It seems that rational functions and exponential function and its inverse -logarithm- continued to complex plane are crucial for defining e and pi proving also e.^(ipi)=-1 and therefore e^(i2pi)=1 which could be also used in the proof of transcendentality of pi. Exponent function and logarithm appear everywhere in mathematics: in group theory for instance.

Certainly exponentiation is fundamental operation in group theory and in integral calculus. Also in p-adic context these functions are very special. e^(px) exists in p-adic number field Q_p for p-adic integers x. Also sin(px) and cos(px). Now e is in special role. Logarithm is the inverse of exponent function and also this is in key role.

To be continued.....

At 9:49 PM, Anonymous said...

Continuation to the response to Santeri's question.....

What is so special in exponentiation? Why it sends algebraic numbers including even integers to "knowably" transcendentals.

a) In Cantor's approach to infinity ordinals exponentiation is involved besides sum and product: all three binary operations -sum, product, exponent are expressible set theoretically. Product and sum are "algebraic" operations. Exponentiation is "non-algebraic" binary operation in terms of powers set: set of subsets. Formation of power set means going to the next level of abstraction: seeing the set of subsets of set instead of set or studying instead of set the set of functions from the set. This could be set theoretic view about transcendence. Could the knowably transcendental numbers like e and pi relate to the transcendence in this set theoretic sense.

b) Maybe exponentiation in set theoretic sense is the mathematical counterpart of transcendence and this is due to the fact that power set means abstraction. In the case of Boolean algebras it means formations of statements about statements. In the case of integers it means formation of non-countable set- going to a new level of infinity.

c) For infinite primes one however replaces set theoretic infinity with purely number theoretic: motivation could be the avoidance of problems like Russell's antinomy: set as its own element. What transcendence could mean as purely number theoretic sense without power set? I have proposed that transcendence as going to the next level in abstraction hierarchy would correspond to second quantization at given level of the hierarchy of infinite primes. The many particle states of given level would become elementary particles of the next level. Could one relate the hierarchy of infinite primes to an hierarchy of transcendentals or is it only infinite hierarchy of algebraic numbers as some arguments suggest?

This leads directly to the question: could one speak about infinite-P p-adicity as I have proposed. Could the infinite-P p-adic fields and corresponding space-time sheets represent mathematical cognition able to deduce analytic formulas in which transcendentals and transcendental functions appear. Group theory would be basic example of this.

d) Giving up set theory in the construction of infinite ordinals would not lead to problems since the proof of transcendentality of exponents of rationals is purely number theoretic as also the proof of transcendentality of pi!

Fascinating question!

At 4:41 AM, Blogger Santeri Satama said...

Thanks Matti,

Your answer led me to the wikipedia article about transcence degree ( where it is revealed, among others, that:

"# Q(√2, π) has transcendence degree 1 over Q because √2 is algebraic while π is transcendental.
# The transcendence degree of C or R over Q is the cardinality of the continuum. (This follows since any element has only countably many algebraic elements over it in Q, since Q is itself countable.)
# The transcendence degree of Q(π, e) over Q is either 1 or 2; the precise answer is unknown because it is not known whether π and e are algebraically independent." (in other words, it is not known if π + e is irrational)

The unknown degree "1 or 2" puts me in the middle of it (1,½) and again makes me bang my forhead in frustration (and hope?) because again, Eulers identity + 1½ directs to the critical line of Riemann hypothesis. Monism or Dualism in the standard Boolean logic -> the Buddhist Middle Way logic of Something Else?

"Many particle state" of transcendality level/hierarchy Q(π, e) of "1 or 2" becoming The "elementary particle" Eulers Identity + 1½ = Riemann hypothesis as axiomatic number theoretical limit or cornerstone for WCW???!!!

WHY is the real part of RH exactly ½ and not something else? If this - I cant say I really know what I'm talking about - is not just a function of my garbled brain and there is a scent of something very deep here, a formal proof of RH with infinite prime and transcendentality hierarchies/levels, p-adics and adeles and quantum boolean would IMHO be Just Lovely. :)

At 11:40 AM, Blogger Santeri Satama said...

I mentioned transcendentals to my father, and his intuition led to the question, is there a "natural" way, analogous to rationals (3 + ½) and complex (a+bi), to represent transcentals (t), e.g. t=rational number(r)+t_2("transcendental unit").

Hmm. Would an unknown transcendental minus a known transcendental be rational? Hard to say and probably this wouldn't lead to much to say.

Next, my fathers intuition lead to pythagorean theorem, where t could be the hypotenousa (square root of r+t_2), t=lim(r+t_2)
t_2 -> 0

My fathers approach seems closely related to


Also the constant problem ( seems very interesting: "Given an expression involving known constants, integration in finite terms, computation of limits, etc., determine if the expression is equal to zero. The constant problem, sometimes also called the identity problem (Richardson 1968) is a very difficult unsolved problem in transcendental number theory. However, it is known that the problem is undecidable if the expression involves oscillatory functions such as sine."

So, in this case sine-wave carries us back to Gödel.

At 11:51 AM, Blogger Santeri Satama said...

Dear Pesla,

Thanks for your thoughts and 90 degrees correction. Thus in yogi terminology real line is the "sushumna", vibrating with imaginary unit and irrantional transcendental constants as ida and pingala - the wibrating wire that does not kill the Schrödingers cat:

At 8:48 PM, Anonymous said...


I woul restrict transcdenence degree to transcendental extensions of rationals since they differ from algebraic extensions. Algebraic extensions are finite-dimensional algebraically but transcendental extensions are infinite-D in this sense since all powers of transcendental are algebraically independent.

Whether to add pi to p-adics is one long standing question: it would extend p-adics to infinite-D number field in algebraic sense since all powers of pi would be needed.

Whether Riemann hypothesis could relate to physics in practical manner is an interesting question. I have seen many proposas.

My own proposal for the strategy of proof relies on a physical argument relying on conformal symmetry which is the basic symmetry of complex analysis.

The argument generalizes a physical argument by Hilbert and Polya saying that zeros are like energy eigenvalues of Hamiltonian: the problem is that the Hamiltonian would not be Hermitian. Hilbert Polya argument is generalized: zeros would be complex numbers characterizing coherent states rather than energies for energy eigenstates.

Algebraic independence can be used in definition of transcendentality. By adding to transcdendental a rational you get a pair of transcendentals whose difference is rational: nothing new;-).

I wrote a posting about transcendentality in TGD framework since at morning walk yesterday I realized that the hierarchy of infinite primes seems to allow to regard algebraic numbers defined at higher levels of hierarchy as transcendentals at the lowest level.

["Seems to X" is for an older statesman a synonym for "X+s" : makes possible to avoid losing face;-)]

Everything would be algebraic and number theoretic in generalizes sense of words- not only physics but also mathematics would be generalized number theory- if one allows the generalization of number concept by bringing in the hierachy of infinite primes and generalization of the notion real number forced by it.

It is really fascinating to realize that foundations of mathematics are not God given as young student of mathematics and physics learns. The foundations reflect our views about reality. In standard set theory based mathematics this reality is that of classical physics!

Mathematician and his cognition is completely absent just as physicist is absent from classical physics and to high degree even from standard quantum theory.

Quantum physics seems to bring in profound modifications to foundations of mathematics: especially so if one tries to understand also the mathematical cognition. A good example about new questions: What in math cognition makes it possible to know that there are transcendentals?

These modificationsx are not visible in practical life in any manner but mean profoundly new vision about physics itself.

At 1:10 PM, Blogger ThePeSla said...


Yes, phase space concepts and even the idea of invariance where the coordinates relate or are independent of things like mass- is no longer able to give us a better and deeper picture of physics- nor for that matter is the quantum theory although as Matti points out it seems to change our view of core mathematics.

I begin to doubt that in these standard terms alone we can ever solve some of these issues we have imposed upon ourselves since Newton.

This is not to say that Matti's ideas are easily seen and to be promoted, nor that they are as simple as some seem to see or believe- they are transcendent and represent the more democratic view of access to learning- so as far as money and recognition go (the Swedish syndrome of the "dudes" notwithstanding) the net is after all the issue of popularity and such virtual reward.

What, I ask you, on the microlevel is such a quantum cat made of and where in the heck is any right angle in the linearity?

The PeSla

At 11:05 PM, Anonymous Trigonometric Equations said...

I am here to discuss a simple definition of Boolean logic as-Boolean logic is a system of symbolic logic which is used in computers.Study of mathematical operations performed on binary variables that can have only two values: true or false. It provides a set of rules called Boolean logic that are indispensable in digital computer-circuit and switching-circuit design.

At 1:33 AM, Anonymous said...

Quantum Boolean algebra allows also superpositions of classical elements. In quantum computations sequences of qubits give similar situation. The new element is fermionic realization. Another new element is that in zero energy ontology one can have superpositions of pairs a-->b identifiable as representations for Boolean rules.

At 1:13 AM, Anonymous hyperbola equation said...

I am here to discuss a simple definition of Boolean logic as-Boolean logic is a system of symbolic logic which is used in computers.Study of mathematical operations performed on binary variables that can have only two values: true or false. It provides a set of rules called Boolean logic that are indispensable in digital computer-circuit and switching-circuit design.


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