This essay reviews: the possible inadequacy of infinity; the definition and associated notions of infinity; the definition and associated notions of the infinitesimal; the lack of usage of infinity in the engineering sciences and the reasons for such disuse; the United States public education failure to provide an alternative to infinity to U.S. society or at least a warning against infinity usage; a third option to personally proving or disproving some proposed concept to and for oneself, to prove instead neither provable nor disprovable, described and defined;
Armed Poorly by a Personal Tool
Many people might very well have considered the concept and notion of infinity, and found something about the idea disappointing, at the least. Infinity cognitive conception within our brains can have a feel inconsistent with itself, inconsistent with personal experience in life, overly irrelative, self-contradictory, negatively recursive, or disturbingly dismissive of true lush and textured complexity worthy of respect.
Persons might have actually at some time chosen to describe to someone else the grandeur and mystery of some feature of nature using infinity in their long ago attempts to convey something of merit and logically compelling. Having employed the infinite, infinitely, or infinity terms as part of a statement meant to convey something beautiful, people might have found brains within heads actually aching a bit inside, from cognitive lurch and conceptual collapse of concepts, immediately after having been cognitively built with the human imagination.
In the following textual illustrations, there is a beginning with a warmup issue regarding the issue of the infinity concept, to alert the reader as to some dangers of the concept, and of course to empower the reader with an alternative concept, and some terminology options for personal thinking usage. Then following that there is a progression toward more formal and rigorous proposal final wording, for eventual peer review and consideration. Following that more tedious set of paragraphs, there is a return to a variety of related infinity issues, in each of which case I merely explain that the logical causality chains and relationships merely do not stand up as rigorous, and with reasons compelling to any writer or reader, as to why not adequate to proof or proven status. This is to further or promote my proposal for new status concerning mathematical and scientific infinity concept professional approach, and is of course a self-impassioned promotional plug, which I feel will not disappoint or dishearten the reader. (Do please in time complete the chapter as a whole, to aid in satisfying those results.) I only see generic beauty to overall and ubiquitous principles of nature and science, even if particular cases or instances of implementation or happenstance can sadly disappoint many among us all as being ugly with detrimental effects.
Astronomically Grand in Size, Count or Degree
A term in usage by some scientists to refer to greater than humongous and greater than vast, and even greater than tremendously huge, is the term astronomically large, great, or grand in number, count, or degree. For an example, I not long ago searched via internet for a rounded-off estimate for the number of atoms of which a typical sample of a nominal pounds-weight person would, to scientific concept and notion, be comprised. When I accounted for my differing personal gross weight, the calculated estimated guess came out to be 12 billion, billion, billion atoms. That’s very much short of the infinite, and different from what the infinite could never, maybe to the reader, never be, yet it does aid the reader and author in placing relative scale in perspective: as a fairly awing grand numerical extent and approximate grand quantitative extent. Extent is the word I’ve chosen in this case, as it applies for quantity, quality, time, space, and many derivative qualities assessable, in extent of greatness or grandness, or in extent of minuteness, at these very existent extremes.
Astronomically grand in numerical count, texture features, contrast features, and slippiness features is the nature of what is sometimes mentally reached for, with no other words to personally select from than the “infinite” term, due to poor education upon each of us. “Astronomically grand”, as used in the preceding text, would have broad all-in-category coverage, with shorter double-wording (as quoted previously in italics) which I consider to maybe be far better, and actually well suited for more limited scopes of description, as behaving aversive of thinking much the term “infinity would not distort and misguide matters far too much so, as usage of the term “infinity” could very well cost each of us, when working mentally with it.
The ideas self-conveyed by thinking through astronomically grand in extent and astronomically minute in extent are bounded and finite, and limited warmly, rather than exacting upon our minds such a toll of very unrelatedly ripped open and rendered moot, and rather than as to be “too” free of nature, at least in real mental brain notion of mind, if not in reality outside human bodies. Preferring against infinity thinking, and performing the mental work to find alternative ways via concepts mentally in subjective opinion, leaves all in nature as finite and relatable, and while plenty greatly open in frontiers, though differing by way of maintaining for the thinker good integrity to refer to the hyper lush about detailed and actual nature, without ripping brain thinking processes apart, which could be viewed as having adverse effects on the quality of life of each infinity and infinitesimal employing thinker.
The proposition of finite defends against irrelativity in conceptual spin-around from potentially producing an ugly feeling case of cognitive balk, or worse, an ugly feeling case of mental logical perfection turned into a disheartening sense of over irrelevant irrelativity, by way of logically collapsing concept-to-same-concept destructive recycling.
Maybe the manner in which factors conceptually “relate” to one another in our mental regard, if maintaining grasp on relational ways consistent with our former and current experiencing of living life, defends against this irrelativity, and maybe primarily by way of always relating or contrasting between at least two (as minimum) cognitive logical mental notions of the manner or way in which some factor or feature relates and contrasts with another differing factor or feature. Maybe a lack of cautionary advise from other persons could stand us in danger of an ugly perfective-thinking very cold and disheartening set of results, if we do not learn a different set of ways to conclude wonderings or considerations on matters, if having only considered one and only one factor by which to relate that which is considered. This two or more categories of relating and of style of personal reaction, could be as foundational as range versus domain in two-dimensional functional plotting, or requiring at least two dimensions of projected imagination or paper/screen surface with which would be necessary to perform any and all modeling visually.
Note: The term “relatively” sometimes does not suit well, though of course matters and factors do relate, in some cases more directly in tight correlation (or co-relation), but always at least ever so slightly indirectly, by basic long-range spatial connectedness real-time, if subsequent near-to-far future interrelating within increasingly large zones is considered, of the activity of all material things (though not all things if the term things more loosely refers to such as activity, actions, interactions, or operative modes or factors).
Note: Also, linear-only myopic Relativity Theory, which I consider to be a misguided interpretation of some truly relevant factors, yet put together poorly, has for many decades tainted the terms: relatively, relate, relativity, relations, relationship, and relatives.
Note: Another set of terms very much of merit, and based on similarity in leftmost six alphabetical symbols within each of that set of similar words, carrying that merit in the spelling and usage and self-mention of each such word, could suit and perform well for each person having interest in mental-modeling word language empowerment. In some obvious by nature usage: “contrastingly,” various factors “contrast” with one another, in how they operate in “contrary” complement, in between varied things. C-o-n-t-r-a-.
Note: The term “compared with” sometimes does not suit well. Comparing is a process, and gets involved with testing for a variety of accountable factors, or involved with experimental method by apparatus, or going through task subtasks to assess, in at least some significant detail. Note: A better termed mental “thought exercise” is by no means a poorly termed “thought experiment“, as mental considerations are not performed by way of experimenting with real objective things, nor by experimental method. Experiencing and experiment with things of relevance outside the human brain (as every person of various ages is regularly experiencing/experimenting with things in life and the portions of his or her body relating to those things in life, ever gaining or refreshing skills), are both based on similarity in leftmost five alphabetical symbols, to the terms: “experiment” and “experiencing”. E-x-p-e-r-.
// Portions or All to be Transferred to Newly Spawned or Other Essays-Chapters //
Proposal of New Axiom of Maybe: No Proof or Disproof of Infinity is Compelling as Valid to Justify Various Dangers of Infinity Concept Scientific Usage
Infinity is tricky. I consider its prevalence in populations to be the result of all human mental visualization systems’ predominancy (maybe so attuned, for personal navigation, personal reward [or satisfaction], and body language assessment, plus for imagination recall of past experiencing, or imaginary imagery creativity based on experience and inexperience both, nicely), and how these our visualization systems intrinsically and unavoidably do preemptively and compellingly suggest “figurative” (what I consider to be bogusly, maybe to the reader) unbounded outward-continuing extent to outer-space, if it actually could be that none such irrelative and ripping concept of infinitely might actually exist beyond some universal spatial realm bound, with alternative finite grandness of spatial maximum extent honored as self-viably quite feasible in all directions, instead, if not wise as taken self-provable nor self-disprovable in the process.
Note: In the contemporary debate over the words “know”, “knowledge”, “known”, and “knowing” – some few yet valued factors about a cosmic spatial realm bound could be known of to maybe personally suit in opinion or unpretending mere belief, if not known (which implies: by way of direct experience, or reality matching of indirect hearsay or read-printed experience to other direct experience to avert extrapolation not at all by necessity validly such following, and in at least some detail) nor knowable. It is a matter for the plenty less sure, in spans of level or degree, thinking to oneself using such terms as: believe, belief, consider, and opinion.
Continuing back in vein: A good introductory analogy which could maybe interest the reader might be the issue whenby physicists try well to explain (to those sensibly placing “social economic” type “demand” on others to explain) concerning the Big Bang Theory and its associated theory of a proposed and debatable – beginning – of time. One might ask “what was before the beginning of time?” A very unusually smart and evasive answer to that is a disagreeing discussion of the logical meaning and merit of words themselves: that if time did not exist at some point in reality, then that is nonsense preemptively loaded: as to merely self-inquire about before, with “before” acting as a time word, invokes necessarily more previous time. If at some stage time did not exist, then neither does the concept of “before,” then neither did reality exist nor humorously, stages of development exist.
A month or a minute before involves a finite amount of “more” time… before. This ironic revelation into the implications of time and space word different natures might be new to many readers, and could offer a potential, yet of course optional and personally selectable/rejectable self-invitation to occasionally pause to consider the implications (the usage of the word “meaning” that does not get blindly doubled up with the usage of “meaning” that instead implies logically worthwhile or having satisfying qualities to it) of words when and if encountering them cognitively coming to mind, whether by self-initiative, or during the course of reading, listening, or conversation.
I scientifically argue that similar term or word realm and merit of weighting, regarding the realm of time issues differing greatly from the realm of spatial issues, holds true when contemplating a possible limit to far off-planet outer space and extent. Thus, when considering bounds or limits of farthest outer space, the words “beyond” and “outside” are feasibly unsuitable and ill guiding tools (to word-only English language term association logic). But the limits of cosmic space (or for time issues: of time repetition in mathematics, as all calculator button presses do involve a finite amount of time) can only be considered using word, term, prose, or language logic, instead of self-allowing for imaginary self-depiction virtual or actual graphically observed visualization logic, a much poorer logical tool.
Mental visualization of the issue seemingly to me will maybe always fail to well respect the issues detrimental to the infinity concept, and maybe could most often, by its weaknesses, instead psychologically suggest perfectionistic certain validity of infinity when maybe in reality it is not nearly that very certain, as the space we are familiar with is akin to luggage space, storage space, closet space, and shapes of subjectively regarded objects, and tracts of expanse such as building interiors, terrain, and sky (not individually regard-able, like objects are to perception). Also, the issue of limits or bounds in cosmic space or extent is maybe so boggling to our visual comparison acumen (due to the utter predominance of our visualization system) that we could get visually stuck, and punt off the pesky subject of infinity entirely.
// Essay Major Revision 1 ongoing Prose Review Quality ends here //
Inner Graphical Imagination as a Stumbling Block and/or Invalid Leap
Our mental imaginary visualization system works exclusively with space and spatial relationships, even when concocting new graphical relation arrangements. To my opinion, from my inability to ever experience beyond this planet and this planet’s wealth of limited scientific measurement, we are as unable to consider absolute spatial limits using mental visualization as is a fish unable to scale a mountain. If the fish only knows water and nearby to water; then our visualization systems only knows space and the spatial. Our mental visualization system, activated away from word well play, seemingly will always reflexively imagine more space (or extent, mathematically) regardless of whether that space or extent truly exists.
I start the clincher with a hint. In the below diagram, the first thing imagined is the entire spatial mental backdrop upon which to draw a proposed limit, after which a personal assessment follows. Thus a mind’s eye has figuratively, or notionally, preemptively and unknowingly imagined the space beyond (the left portion plus the right portion of the spatial mental backdrop, depicted below) before laying out the question of limits, thus insuring (upon repetition) the seeming (but often wondered about, in high school classrooms) conclusion of infinity.
Thus, to any given reader, imaginary exercises are possibly false preemptive conclusion based solely on visual psychology and physical brain anatomy, of a species with no evolution near the Cosmic Outer Bound: we humans.
Note: Maybe even in a far off corner beyond an outer galaxy, no microbe or more would reasonably to our notions have experience at the non-edge, but actually more fairly termed bound. The term boundary in common English language usage (rather than somewhat rigid and often debate-prone definition by commerce organizations providing dictionaries in print or internet online) refers very tightly to a demarcation between two zones spatially, and also does the term edge do the same, yet while proposing by referring, to material of spatial shape and form abutting to atmospheric or vacuum space or spaces. The term bound does much better for our mental modeling, affecting a thinker internally and personally in its impact on subsequent thinking, rather than me attending to issues of conversational effect on others socially. Also, the term bounded conveys to the self the notions of limited and constrained, whereas a boundary is between two zones, spatially.
I consider it “false to conclude proven infinity, and also false to conclude proven noninfinite” as a recommendation to any person having interest in the infinity issue, professionally or curiously, to maybe personally adopt. There is great benefit to placing and firming up by standards, a significant dose of healthy mental doubt about infinity either way, the fruits of which might include less occasions of punting to infinity by professionals, and even more so, less occasions of terminal blocks to newly spawned considerations by professionals in whatever profession, job, or vocation from which persons might have interest in anything that relates to any type of scientific matter. This includes nearly every project manager and project guiding and associated personnel, for any commercial trade or governmental organization, regardless of scientific education level.
To disprove infinity would mean to “prove the opposite case must be”, and to my recommendation should not be adopted by a person considering this. This unthinking quick disproval attempt promotes and fosters only the missing of the opportunity and clouding of the value of debating infinity, to the point – or degree or level – of cancellation from the start.
Word logic (which contrasts phrases of clause[s] rather than individual words most often, with special words like infinity most often having meaning full of words to self-present and self-consider, if fairly self-developed in prose) is very disagreeable and unsatisfactory to me, as such an adoption of that sort of conclusion by others I consider to place at great risk the personally adoptable proof of: infinity is not concludable as to provable or disprovable, both, and should be considered to stand in standardized procedural reserved doubt, necessary deprecation, and scientific disregard, with no attempt to change that status during scientific considerations or projects, as to applicability and suitability of the infinity concept to any scientific task or undertaking. This sort of informal or formalized personal policy, without editing for conciseness increase, which would carry an inverse decrease in corresponding elucidation, is as written imbued with discussion topic prompters, and does not condense well. If the concise rule ever applies, it applies only for headers and titles, rather than for this sort of statement.
Small as well as Big
The same applies to our sense of the infinitely small. Our visualization system is rendered out of its element, to be able to consider that the concepts of “inside” or “within” might not apply at some limit.
No Provable Perfectionistic Absolute Zero
Zero is tricky as well. There is zero as a matter of convention, and there is obsessively zero. The small (enough for working models) middle span, between “to the left of” (negative) and “to the right of” (positive) is good and true “reference” zero. Lessening by an incrementally positive amount (subtraction) and greatening by an increment of positively present amount (addition) both deal with similar widgets or fractions of widgets, but + indicates addition, whereas by “convention” minus means in the other direction, or to the left, or below (depending on viewpoint). It is similar with north and south. Real numbers reflect real positive in “presence” relative amounts of widgets, or portions of widgets, or of metrics of their qualities. A negative sign in front of a positive in presence number, is merely the convention: of to the right or to the left, above or below, or greatening or lessening. Whole numbers have to do with the recognizable and/or functional identities of fairly individual things. Items for sale are that way.
// To be continued and/or expanded //
We all could come to soon know that a (0,0) spot on graph paper or on screen depiction, depending on scope and purposes could be viewed in some few cases, as subjective: we could place the axes anywhere we want on the graph paper, with the same shape or line maintaining its nature.
(Functions are a convenient way to have shape in one dimension tied tightly to another. But little in life is slaved that way.)
// To be continued and/or expanded //
Also, the 0 mark at the starting line of a foot race could be viewed, in widest particular assessment criteria, as subjective: we could start the race somewhere else, after all. But what happens when you have 0 apples in your house? Isn’t that an absolute or perfect 0? Nope. The 0 merely reflects that you have reached the lower limit of your category of containment, which in this case is defined by “in my house uneaten” or “in my possession uneaten”. Nature cares for every little detail and does not necessarily care so much for our sense of possession, containment, or mentally frozen in time: form. Thus Nature cares for the core in your trash and the remainder of the apple in your digestive tract, plus the sucrose in the brain. And Nature cares for all the apples in the grocery store or on all the trees, even before you buy some.
Proposal of New Status: There is No Provable Infinite Precision
Just as I don’t conclude in infinity or the infinitesimal, and just as I believe (of which term means less than conclusive…) that texture, flaw, and uniqueness apply to everything, I also do not conclude toward infinite precision. Thus, to me, the number 1 does not in any way imply 1.000… with infinite zeros beyond the decimal point. It might manifest as 1.15s; it might manifest as 1.0005s; and it might manifest as 0.99523s. I add the “s” to denote “smeary” or “slippy” beyond the decimal place of chosen working precision. That slip maybe indicates several things. First, we don’t know or care what follows it. Second, infinite zeros don’t follow it: not necessarily to logical causality. Third, all is in flux with change over time (great or small) and thus at some decimal place down the line there is no specific number, but rather a indicator of “unknown and not of further interest) concerning numbers. Thus in reality, all numbers behave like pi and phi, which I call “defiant numbers”. They defy our attempts at infinite precision and should teach us something about all numbers. By logical negation, a reader of age and experience could assert that to their life: no one has lived long enough to perform infinitely long in time: sequence building operations. An (not truly) infinite loop in computer code would in time be foiled by computer failure, power switch oops, unplugging much later to move to a new home, or not possessing the endurance to make it to the year 3000, to get any “when” near infinite in duration.
The thought exercise (and no experiment) I use here is to think of the googolplex of many decimal places of zeros to the left of the decimal point. What is a googolplex? It’s more fun to search for it online. If that quantity or quality (using numbers) is approached, then it runs away like a googolplex ^ googolplex. If that is approached, then a googolplex (to the googolplex – 1 of power operations in between each of a (googolplex of googolplexes) would be the next figurative sequence of untestable and unexperienced fiction operations (which are good things, or there would be no fun books to read) that might next ensue. And so on. This only took less than a minute to mentally do, to keep on growing ridiculously. This repeated, at the same rate for a thousand years onward, could still not “approach” infinity without it running infinitely more far away. And that many years is far short of infinity in duration. There is no way to prove infinity and also there is no way to (or need to try) to disprove it. I merely call into doubt the “logical necessity” of it. The heart of the problem, would be, if infinitesimal was real, then every cotton ball, could fit a pica-universe inside, and each such universe, could have one cotton ball, to zoom in so far as to have mini-divisible room, to fit another universe of matching texture/detail within, on and on, in unbounded degree. It is a word logic dangerous thing to consider “proven” or “de facto.” It is better to leave it open, and marked as debatable and unprovable and also not disprovable, as no exper-ience of mankind, known as exper-iment, could ever journey that far away, or that afar of future by way of skipping ahead, or that small. That I find, for the reader to independently decide, if self-provable to yourself, the person who experiences your living of life, by negation. (Has anyone ever done it, to your self’s notice?)
<– More to come on a proposed existence of a Planck-like “Smear-limited timespan,” with no aim to define it in too precise a particular number, but rather just to suggest maybe its presence continually, and of course considering relative postulated realm, or regime, in numerical time span maximums and minimums bounding, likely. –>
// To be continued and/or expanded //
Half by Half taken to Infinity
Thus the maybe silly problem of a man traveling to a location by always stepping half of the remaining distance is foiled by Nature. To my considering, Nature will not allow anyone, any particle or any wave to step too exactly, at a perfective 0.5000… with infinite zeros, even for one step, much less for ensuing never-ending half-operations on the distance previously stepped. Even one such step, by any noun-referable thing or verb-referable interplay betwixt things, would by ridiculous necessity (a prompt for a giggle, in eventual potential reaction) involve cutting the universe into two with an infinitely sharp demarcation. This I would not attempt, regardless of footwear.
To be fair to those who could disagree, even if Nature did allow this, then at some point the Planck length would foil it, and pop across the finish line to terminate the bogus (to me) concept of never-ending. Consider that if valid, then the Planck length would permeate everything around us and imply a minimum grain size of spatial extent in three dimensions, as well as imply that each atom of the Periodic Table of Elements reference could actually have portions of interiors to each unique copy of such similar atom categories to which farther inward spatial divisibility just does not apply. Additionally and very telling, merely because the Planck length is proposed to be so very small, doesn’t change the position that it would pertain everywhere within a common item such as a house construction or remodeling two-by-four spar. A way or means to explain this is that if such minimum spatial extent is valid, or if unprovable is compelling to model with, then it would thus affective at all scales, by accumulation principle.
Minimum Interior Inside of which Spatial Sub-divisibility Might not Apply
To textually illustrate the Planck length idea, to start it is merely proposed as a reference length, inside of which no sub-grid of graph paper style is proposed to be sensible, or necessarily provable. One could visualize it like a pill capsule, with fuzzy edges, with race track emptiness in the middle, with the trick of overlaying tiny graph paper sub-divisible backing lines all over the diagram, except for within the empty-of-grid reference zone (spatially truly empty is actually nonsensical in this concept scenario, as empty is a space-laden term also). An annotation arrow could point functionally toward that non-empty non-zone, and annotate some proposals, or mere postulates, and no proof, of words of logical merit. Maybe the reader could simply perform an imaginary cut-and-paste of something written herein, or have more to write.
Experiment, Modeling, and Measurement
Proof can only be made with experience, whether using mensuration by numerical systems or just eyeballed with real brain view via eyes, using relative size sticks, unmarked, for a case to consider. Experiment just observes the experience of things, as its basis, and though often most conveniently managed with measurements, does not require that, especially to the science of children assessing toy performance with those five senses. Modeling is more complex, and is used to derive wisdoms from what has been already experimented with experientially. The term “measurement” implies numerical accounting of quantity or quality.
Limitations are Real and Do Follow well from Logical Considerations
At some scale smaller than that which electron microscopes can observe, seems to me beyond our scale, and thus beyond our experience. The idea of possible device designs small enough to observe molecules in action I scientifically argue would by necessity involve device surfaces composed of several thousands of degrees more molecules of Iron and Silicon and more, to attempt to measure something as small and nebulous to us in graphical depictable nature and in behavior as another solo or small bundle of molecules or set of atoms. Such a device tool would have great difficulty in maintaining shape and observing through atmospheric Dioxygen and Dinitrogen, especially. Thus I wonder that electron microscope observations could be the limit after which pushing smaller only results in excessive failure to depict accurately or to hold shape with stability, rendering it impractical.
Poorly Conceived Bad Sells
In contrast, nuclear engineering has experimental results very well proved by way of thermonuclear power plant effectiveness and thermonuclear weapon effectiveness and viability. The body of knowledge distributed in varied personal versions among the members of the field of that sort of engineering could stand as better sources of closer to true, at least seemingly, ideas and wisdoms concerning behavior of atoms and molecules. Interestingly, I have read that to nuclear engineering, the only compelling to the student or reader suggestion as to amounts of timespans involved as posed concerning our sun star, is a rounded to the nearest billion years rough gouge of not more than about five billion years before presently, as the idea and notion (and no more) of maybe when our sun began forming from a previous star’s having aged and ceased fusion operations. Any extrapolation to before that reference amount of time before presently, is up to conjecture and fictional modeling, via thought exercise only.
Maybe also do consider that the other details published, or marketed, concerning nova and supernova behaviors, could use some scrutiny as to whether based in experiment, or wantonly extrapolated to the extreme of movie theatre fiction. I do not know, and presently I have not a way to discover such a discussion, which might retreat off of too much ostensibly-certain (while not at all) non-truths about how stars form and terminate. This sort of presentation and delivery the reader could maybe have noticed in many cases before, concerning many factors to many things so often presented as the perfect knowledge, with those persons involved in purveying such dross seemingly unaware that it truly does no one any better that way, failing to satisfy as seemingly not realistic. To me, I term that these as “poorly conceived bad sells.”
No human can shrink in size to as small as an atom, is the posed postulate. That could maybe be proven by negation, if noticing it fail at every attempt by relatives, family, friends, acquaintances, and even unfamiliar other persons. If some feat has never been observed, then that does not prove it impossible, but it adequately scores sure enough in level of less than known degree of surety, for many persons to be open to the opinion that such unobserved and seemingly maybe unobservable phenomenon do not stand as compelling for belief in, nor likely wise personal investments for modeling usage, to any relevant degree.
// To be continued and/or expanded //
// To be adjusted for smoother delivery of concepts, spelling and grammar, untightening of dense packaging of concepts - done initially to begin with coherence among multiple factors, better readability, and/or better tone of demeanor and mood at implied benefits and implied beauties of nature: human - machine - suburban - and wilderness natures. //
with those total personal neurons of finite total estimated by research and forensics to be somewhere near 120,000,000,000 or one hundred and twenty billion, in each human brain of adult size.