Rob
I'm going to address this a number of different ways.
1. Computer Random Number Generators
First of all, none of the online tarot sites use random number generators that are truly "random." All of these sites use a mathematical formula based on a time seed to generate pseudo-random numbers which correspond to tarot cards. While some of these algorithms are fairly good, at best they are equivalent to "shuffling a lot."
For those that believe the feeling that tells you when to "stop" shuffling the cards and draw, or the feeling that tells you to pick a particular card in a fan, is governed by some internal or external "higher power" (be it a higher self, a collective unconscious, a personal deity, etc.), there's no reason that said force couldn't give you that same feeling about exactly when to click the mouse on the "Draw" or "Shuffle" button that coincides with drawing the cards you need online. With a time-seeded random number generator, in fact, the act of shuffling cards by hand and stopping at a particular point is precisely equivalent to clicking a button to use the state of the randomizer at a particular moment in time - the only difference is that the cards are being effectively "shuffled" much, much faster.
By the same token, the same "feeling" might guide your actions in whether you use the meanings that come up on the screen or read the cards for yourself. In either case, you're getting the message you're "supposed" to get, regardless of whence you believe that message actually comes. One might argue that your actions are "accounted for" by this higher power, whatever it is, in selecting which cards you get.
There exist random number generators that use atmospheric noise, quantum radiation, and other very good ways of generating near-random numbers, but none of these are used by what are ostensibly "entertainment" websites. Even those may not be "truly random" if the universe is fundamentally governed by equations and contains no random element, which is a concept still debated by modern scientists and theologians alike. My inclination, though, is that in this case it doesn't matter - provided you are "attuned" to whatever force compels you to act in a way that results in getting the cards you should, it shouldn't matter how the cards are randomized.
2. How You Connect with the Cards
Many of you probably do some basic routine things to prepare for a reading - perhaps grounding and centering, meditating, Reiki, perhaps setting an atmosphere. The purpose of these rituals is essentially to attune yourself to that power, be it a heightened state of awareness, a shamanistic trance, or a connection with a collective unconscious or personal deity. For the less spiritually inclined, it would simply serve to focus your concentration and prepare you for the task at hand. Do you do those things before generating a point-and-click reading? I would wager not, in most cases. Would you expect your own physical cards to "work" if you just took them out and threw them down with no preamble, half-expecting them not to work, as most people do with online readings? Certainly not. I would argue the same is true with online readings.
And even if you don't subscribe to any of the spiritual angles to Tarot, there are other purely psychological reasons why online readings may not work for you. For those who are not computer experts, the mild stress incurred by the difficulty of using the medium may be distracting or otherwise divert attention away from interpreting the cards. Even if you are an expert, typically one multitasks while using a computer, and it's easy to be distracted by other things onscreen. Even the very activity of using a computer can induce strong feelings of impersonality that are difficult to shake, and these can also interfere with one's ability to concentrate on the meanings of the cards on a purely intellectual level.
3. What You Can Do About It
(a). Do your usual reading routine and prepare yourself psychologically and emotionally prior to doing the web reading, just as you would for a reading in person. This will ensure that you're at full intellectual capacity, relaxed, motivated, concentrating on the task at hand, and if you believe in it, connected to your higher self or spiritual force of your choosing.
(b). Do (a), then do the reading on the web...but instead of interpreting it on screen, use the reading that came up and lay out the cards in real life, THEN interpret it. You might say it defeats the purpose, and it does, but you can use this purely as an exercise to do two things, (i) improve your confidence in the online medium by proving to yourself that the reading generated onscreen is valid and relevant (since I would wager that, with the physical cards on the table, you'll be in your usual element and be able to make sense of the cards you drew as though you'd drawn them by hand), and (ii) forge an automatic connection in your mind between the images of the cards in your favorite online reader and the mental image of those cards sitting on your reading table. This is very similar to the mechanism that makes it difficult for people who have always used one Tarot system (RWS, for instance) to read a new system (like Marseilles or Thoth) - they always revert back in their heads to the familiar interpretations from the system they've used extensively. In this case, however, the same style of mental association is beneficial - it bridges the gap between seeing cards on screen and seeing them on your reading table, and that should help you draw upon your interpretation skills more effectively.
I hope that I've introduced some clarity to the issue at hand through the above discussion and not hopelessly muddled it in a sea of superfluous verbiage. Just some thoughts!
1. Computer Random Number Generators
First of all, none of the online tarot sites use random number generators that are truly "random." All of these sites use a mathematical formula based on a time seed to generate pseudo-random numbers which correspond to tarot cards. While some of these algorithms are fairly good, at best they are equivalent to "shuffling a lot."
For those that believe the feeling that tells you when to "stop" shuffling the cards and draw, or the feeling that tells you to pick a particular card in a fan, is governed by some internal or external "higher power" (be it a higher self, a collective unconscious, a personal deity, etc.), there's no reason that said force couldn't give you that same feeling about exactly when to click the mouse on the "Draw" or "Shuffle" button that coincides with drawing the cards you need online. With a time-seeded random number generator, in fact, the act of shuffling cards by hand and stopping at a particular point is precisely equivalent to clicking a button to use the state of the randomizer at a particular moment in time - the only difference is that the cards are being effectively "shuffled" much, much faster.
By the same token, the same "feeling" might guide your actions in whether you use the meanings that come up on the screen or read the cards for yourself. In either case, you're getting the message you're "supposed" to get, regardless of whence you believe that message actually comes. One might argue that your actions are "accounted for" by this higher power, whatever it is, in selecting which cards you get.
There exist random number generators that use atmospheric noise, quantum radiation, and other very good ways of generating near-random numbers, but none of these are used by what are ostensibly "entertainment" websites. Even those may not be "truly random" if the universe is fundamentally governed by equations and contains no random element, which is a concept still debated by modern scientists and theologians alike. My inclination, though, is that in this case it doesn't matter - provided you are "attuned" to whatever force compels you to act in a way that results in getting the cards you should, it shouldn't matter how the cards are randomized.
2. How You Connect with the Cards
Many of you probably do some basic routine things to prepare for a reading - perhaps grounding and centering, meditating, Reiki, perhaps setting an atmosphere. The purpose of these rituals is essentially to attune yourself to that power, be it a heightened state of awareness, a shamanistic trance, or a connection with a collective unconscious or personal deity. For the less spiritually inclined, it would simply serve to focus your concentration and prepare you for the task at hand. Do you do those things before generating a point-and-click reading? I would wager not, in most cases. Would you expect your own physical cards to "work" if you just took them out and threw them down with no preamble, half-expecting them not to work, as most people do with online readings? Certainly not. I would argue the same is true with online readings.
And even if you don't subscribe to any of the spiritual angles to Tarot, there are other purely psychological reasons why online readings may not work for you. For those who are not computer experts, the mild stress incurred by the difficulty of using the medium may be distracting or otherwise divert attention away from interpreting the cards. Even if you are an expert, typically one multitasks while using a computer, and it's easy to be distracted by other things onscreen. Even the very activity of using a computer can induce strong feelings of impersonality that are difficult to shake, and these can also interfere with one's ability to concentrate on the meanings of the cards on a purely intellectual level.
3. What You Can Do About It
(a). Do your usual reading routine and prepare yourself psychologically and emotionally prior to doing the web reading, just as you would for a reading in person. This will ensure that you're at full intellectual capacity, relaxed, motivated, concentrating on the task at hand, and if you believe in it, connected to your higher self or spiritual force of your choosing.
(b). Do (a), then do the reading on the web...but instead of interpreting it on screen, use the reading that came up and lay out the cards in real life, THEN interpret it. You might say it defeats the purpose, and it does, but you can use this purely as an exercise to do two things, (i) improve your confidence in the online medium by proving to yourself that the reading generated onscreen is valid and relevant (since I would wager that, with the physical cards on the table, you'll be in your usual element and be able to make sense of the cards you drew as though you'd drawn them by hand), and (ii) forge an automatic connection in your mind between the images of the cards in your favorite online reader and the mental image of those cards sitting on your reading table. This is very similar to the mechanism that makes it difficult for people who have always used one Tarot system (RWS, for instance) to read a new system (like Marseilles or Thoth) - they always revert back in their heads to the familiar interpretations from the system they've used extensively. In this case, however, the same style of mental association is beneficial - it bridges the gap between seeing cards on screen and seeing them on your reading table, and that should help you draw upon your interpretation skills more effectively.
I hope that I've introduced some clarity to the issue at hand through the above discussion and not hopelessly muddled it in a sea of superfluous verbiage. Just some thoughts!