The Boltzmann Paradox isn't really a paradox (in my opinion), but more of an argument for solipsism: It starts off by talking about the probability of a single self-aware brain forming in the universe (let's call this probability P(brain)). Then, it points out that from a naive inspection of the world around us, there seems to be at least 6 billion self-aware brains in the universe, and perhaps much, much more than that. It then compares the probability of there truly being billions of self-aware brain versus the probability that there is only a single brain merely hallucinating the existence of billions of other brains, and finds the latter to be of much higher probability.

At first, I found this argument seductive: Consider the chance of rolling "1" on a 6 sided die. The probability of this is 1 out of 6, or approximately 17%. What's the probability of rolling two dice and having them both come up 1? It's the previous probability taken to the power of 2, which is 1 out of 36, or about 3%. What about for 3 dice? 1 out of 216, or about 0.5% (half of 1%). What about the chance for six billion dice? Well, I tried to write a computer program to calculate what the probability of that would be, but it took too long to run, so instead, I'll give you the probability for six thousand dice. First, I'll give you the number of digits in the probability for six thousand dice. The probability is 1 out of some number, where that number has 4669 digits. Note that the number "google" merely has 101 digits. This number is way bigger than a google. The number itself, I'll write in its own paragraph:

8 081 692 125 860 371 891 061 517 209 327 769 103 574 833 821 732 836 132 378 031 121 712 370 334 199 206 523 140 452 904 728 965 008 647 823 843 880 057 347 947 370 727 240 942 428 033 162 495 520 757 520 782 294 982 606 468 677 382 429 911 936 409 094 094 916 642 815 351 222 219 746 530 740 948 024 145 885 556 214 987 602 801 486 860 782 903 312 324 244 939 647 130 343 726 092 246 298 511 180 421 015 121 744 609 527 969 444 825 561 596 736 513 921 103 520 298 940 880 940 407 476 518 150 496 965 931 482 485 801 525 364 935 569 809 329 146 391 072 032 775 764 105 365 875 493 656 593 491 608 368 500 546 163 301 778 134 084 227 650 475 894 005 217 276 105 003 941 897 015 075 473 724 768 127 173 687 622 461 217 545 731 890 508 376 566 758 825 157 263 261 693 578 937 674 503 532 275 835 140 908 095 939 829 132 043 249 859 224 338 026 803 023 727 471 928 694 894 996 369 138 568 657 674 898 308 384 333 712 800 218 800 396 374 370 040 620 796 629 027 026 567 461 955 014 435 526 063 194 992 914 918 378 046 417 278 009 903 936 733 107 172 542 261 582 100 513 776 254 426 176 423 067 707 194 479 379 369 387 219 370 899 045 339 346 845 036 905 191 054 411 873 969 177 399 671 132 627 671 895 911 451 255 006 539 430 750 007 217 863 040 693 268 122 564 117 981 863 281 860 346 231 062 640 837 427 764 960 323 421 717 742 697 770 057 335 323 313 831 115 203 241 038 814 269 818 244 312 117 740 467 146 295 790 840 251 332 239 812 098 678 711 460 386 996 817 679 344 710 397 130 105 692 568 376 649 410 944 086 334 754 941 812 207 461 556 100 942 097 028 358 137 335 113 636 855 662 064 826 095 821 899 132 179 598 430 765 656 045 628 756 674 118 049 037 063 070 366 027 201 763 276 344 745 216 723 725 756 023 476 660 272 596 362 745 924 240 090 469 672 117 035 946 048 108 495 942 149 160 463 905 106 321 949 968 591 556 130 466 380 528 448 784 610 047 183 983 733 179 405 380 470 695 494 652 606 656 977 419 357 483 963 623 334 325 564 492 940 188 938 522 133 679 356 445 687 201 908 339 936 768 525 080 999 221 516 105 025 541 346 700 289 403 793 482 096 578 627 203 206 614 491 706 247 443 033 156 577 252 250 029 747 793 800 915 538 558 638 885 200 258 061 806 157 812 899 332 667 694 945 280 541 676 544 382 185 879 527 475 965 829 612 879 586 701 900 514 026 706 730 700 673 655 757 062 984 967 612 544 656 955 657 778 997 801 139 402 456 614 697 639 370 992 035 160 099 424 461 288 019 935 847 430 835 657 368 696 076 500 838 581 103 758 764 001 732 932 807 164 520 487 154 317 937 833 109 707 738 518 109 993 554 850 318 949 796 860 514 145 072 016 570 302 806 331 269 914 186 195 599 889 739 683 429 235 323 388 552 339 214 718 426 732 349 215 274 798 958 739 646 985 588 800 015 414 227 420 488 847 832 082 322 554 700 731 763 994 370 398 474 072 980 481 226 233 614 655 631 623 389 005 801 347 183 865 442 961 906 542 773 707 492 639 746 941 194 251 152 429 243 924 222 861 326 385 878 989 046 547 240 392 113 773 563 950 203 745 411 479 740 812 749 422 280 655 801 698 070 229 880 392 282 543 811 670 823 079 874 440 812 175 316 631 029 566 153 299 754 886 484 765 907 272 937 734 460 022 135 632 339 375 453 398 821 676 488 704 776 155 935 783 089 007 782 953 618 765 465 989 636 147 648 781 627 501 027 453 927 838 136 901 633 214 445 645 284 808 583 396 287 851 423 519 230 782 881 280 142 421 682 621 016 017 699 605 918 226 101 524 395 858 126 512 909 340 260 342 476 842 733 255 069 833 782 508 035 753 056 334 727 717 193 413 560 621 136 093 647 247 535 411 363 950 914 060 040 981 701 751 341 052 691 406 369 192 961 371 479 896 165 008 903 763 331 270 161 054 093 701 132 646 766 166 431 172 580 081 470 655 888 780 903 422 122 237 368 809 538 905 233 040 343 866 737 327 966 400 619 022 314 528 325 110 237 137 605 185 673 281 962 415 097 484 581 212 919 109 809 688 815 217 329 517 155 191 138 048 692 766 847 682 818 992 554 842 834 565 843 529 417 112 808 904 983 513 839 902 889 041 559 828 179 809 722 717 444 933 119 215 774 040 327 379 979 499 058 736 568 784 044 577 009 953 844 227 330 016 322 704 515 432 608 225 723 114 302 777 054 296 103 383 595 172 975 219 393 294 002 832 298 578 610 726 765 355 481 676 019 130 535 752 805 823 625 751 097 051 380 632 076 451 747 480 646 178 645 631 470 609 810 618 996 819 651 380 869 119 546 394 079 300 414 689 913 869 447 300 885 023 233 333 451 392 591 036 716 173 889 441 733 795 544 513 255 247 980 555 339 383 402 716 332 621 667 756 954 124 764 786 983 443 738 382 653 745 765 455 553 957 962 274 751 442 246 940 740 816 846 002 024 841 616 269 189 897 202 108 171 366 816 152 347 043 451 187 729 664 894 740 623 003 670 423 555 121 150 827 638 425 734 865 003 648 623 756 225 399 970 916 045 042 097 714 539 539 867 315 290 331 647 113 740 408 272 641 096 000 733 544 907 372 985 055 928 608 483 370 505 025 464 295 222 713 186 863 168 074 408 606 353 060 500 375 828 286 400 072 401 960 749 737 084 946 185 482 321 886 274 300 436 858 801 290 183 405 174 723 267 237 420 278 379 571 234 301 472 595 675 122 072 442 564 870 348 216 391 223 326 466 738 161 216 153 069 206 553 377 408 115 173 387 949 285 196 692 637 446 001 494 078 044 655 654 939 781 029 746 611 151 058 530 736 874 592 572 235 968 389 629 081 757 362 575 414 177 598 955 646 441 261 609 941 320 974 861 337 213 277 205 045 486 877 099 194 520 588 714 062 034 935 837 275 897 664 283 455 756 074 890 545 636 223 512 589 547 721 773 380 167 515 172 728 219 262 537 310 106 901 439 271 130 151 589 873 190 541 741 854 168 175 028 509 128 056 477 092 785 291 299 746 540 202 708 361 454 612 190 821 225 686 479 660 717 909 594 985 110 499 426 547 372 491 036 111 819 485 668 253 788 198 056 098 136 653 934 714 215 138 226 833 662 071 960 488 524 373 796 192 624 675 683 159 057 480 546 290 673 986 120 545 154 165 570 601 554 611 866 397 884 433 857 686 064 253 095 521 871 236 122 234 651 167 966 323 694 134 748 513 546 707 455 426 824 972 203 868 670 531 971 746 291 249 919 611 839 819 826 386 756 042 529 887 668 125 128 750 588 197 323 466 123 013 754 750 413 868 061 555 357 302 258 603 838 984 959 505 852 831 574 673 785 044 396 888 164 528 146 402 130 407 751 990 570 256 187 475 534 063 472 842 717 965 826 667 490 669 238 264 484 955 176 411 162 623 529 189 376

So that's merely the probability for six thousand dice, a number so large, it's beyond my ability to even try to visualize in any way. Imagine what the probability must be for six million, or six billion dice.

So coming back to the Boltzmann Paradox, we have the probability of a single mind forming in the universe as being some value (perhaps it's 1 in 6, perhaps it's something smaller, perhaps it's something larger). No matter what that value is, the probability of six billion brains forming in the universe must be something absurdly much more improbable. It would seem that it's almost guaranteed that solipsism is true, since it is practically infinitely more probable that there is only one mind, hallucinating about the existence of the universe as a whole, and the existence of other minds, than there actually, truly are, other minds out there.

So I was convinced by the Boltzmann Paradox for a day or two. Then I realized a flaw in the argument.

The argument assumes that each subsequent event is independent from the previous event. This is true in the case of dice (the fact that you rolled "1" on the first die has no effect on the chances of rolling "1" on the next die), but I don't think it's true for the formation of human minds. We all know at least one mind exists, thanks to Descartes's "I think, therefore I am". Now there are many ways for a mind to come into existence, some more probable than others. It's possible, for example, that random particles in the air just happened to coincidentally collide in each other in the perfect way to spontaneously for a mind with "false" memories of the past. Another possibility is that minds came into existence gradually via evolution. Because I'm too lazy to demonstrate it via argument, I'm going to take as an axiom, that the "evolution" possibility is much more likely than the "spontaneous mind formation" possibility. And in fact, I'm going to just take as a temporary axiom that evolution is the most likely possibility (and later on in this post, I'll look at what happens if we remove this axiom).

So again, we know at least one brain exists, and from this, we can deduce from this that evolution probably occurred, because we're assuming that it is the most likely explanation for the formation of that single mind in the first place. Well, if evolution probably occurred, then other minds probably also exist. This is what I mean by the formation of minds not being even independent with itself: If we have at least one mind (and if we assume it arose due to evolution), then we probably have several other minds as well. This is where Boltzmann's solipsistic argument falls apart. The chances of there being one mind hallucinating the existence of other minds is no longer much more probable than the existence of billions of other minds.

So what happens if we remove the assumption that evolution was the most likely way for brains to form? Well, my arguments still stand, as long as the most likely way for brains to form encourages the formation of several brains versus one. So the "spontaneous formation" theory, for example, doesn't follow this condition, because if the spontaneous formation theory were true, there would probably be only one brain, and not many. The evolution theory does follow this condition, because if we had one brain come into existence due to evolution, we probably have many brains. Intelligent design also adheres to this condition: if an intelligent designer created on brain, there's a good possibility it created several other brains as well.